YES

We show the termination of the TRS R:

  active(zeros()) -> mark(cons(|0|(),zeros()))
  active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
  active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
  active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
  active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
  active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
  active(U106(tt())) -> mark(tt())
  active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
  active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
  active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
  active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
  active(U114(tt(),L)) -> mark(s(length(L)))
  active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
  active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
  active(U122(tt())) -> mark(nil())
  active(U13(tt())) -> mark(tt())
  active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
  active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
  active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
  active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
  active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
  active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
  active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
  active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
  active(U23(tt())) -> mark(tt())
  active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
  active(U32(tt(),V)) -> mark(U33(isNatList(V)))
  active(U33(tt())) -> mark(tt())
  active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
  active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
  active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
  active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
  active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
  active(U46(tt())) -> mark(tt())
  active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
  active(U52(tt())) -> mark(tt())
  active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
  active(U62(tt())) -> mark(tt())
  active(U71(tt())) -> mark(tt())
  active(U81(tt())) -> mark(tt())
  active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
  active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
  active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
  active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
  active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
  active(U96(tt())) -> mark(tt())
  active(isNat(|0|())) -> mark(tt())
  active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
  active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
  active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
  active(isNatIList(zeros())) -> mark(tt())
  active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
  active(isNatIListKind(nil())) -> mark(tt())
  active(isNatIListKind(zeros())) -> mark(tt())
  active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
  active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
  active(isNatKind(|0|())) -> mark(tt())
  active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
  active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
  active(isNatList(nil())) -> mark(tt())
  active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
  active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
  active(length(nil())) -> mark(|0|())
  active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
  active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
  active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
  mark(zeros()) -> active(zeros())
  mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
  mark(|0|()) -> active(|0|())
  mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
  mark(tt()) -> active(tt())
  mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
  mark(isNatKind(X)) -> active(isNatKind(X))
  mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
  mark(isNatIListKind(X)) -> active(isNatIListKind(X))
  mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
  mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
  mark(isNat(X)) -> active(isNat(X))
  mark(U106(X)) -> active(U106(mark(X)))
  mark(isNatIList(X)) -> active(isNatIList(X))
  mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
  mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
  mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
  mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
  mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
  mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
  mark(s(X)) -> active(s(mark(X)))
  mark(length(X)) -> active(length(mark(X)))
  mark(U13(X)) -> active(U13(mark(X)))
  mark(isNatList(X)) -> active(isNatList(X))
  mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
  mark(U122(X)) -> active(U122(mark(X)))
  mark(nil()) -> active(nil())
  mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
  mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
  mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
  mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
  mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
  mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
  mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
  mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
  mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
  mark(U23(X)) -> active(U23(mark(X)))
  mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
  mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
  mark(U33(X)) -> active(U33(mark(X)))
  mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
  mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
  mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
  mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
  mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
  mark(U46(X)) -> active(U46(mark(X)))
  mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
  mark(U52(X)) -> active(U52(mark(X)))
  mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
  mark(U62(X)) -> active(U62(mark(X)))
  mark(U71(X)) -> active(U71(mark(X)))
  mark(U81(X)) -> active(U81(mark(X)))
  mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
  mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
  mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
  mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
  mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
  mark(U96(X)) -> active(U96(mark(X)))
  cons(mark(X1),X2) -> cons(X1,X2)
  cons(X1,mark(X2)) -> cons(X1,X2)
  cons(active(X1),X2) -> cons(X1,X2)
  cons(X1,active(X2)) -> cons(X1,X2)
  U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
  U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
  U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
  U101(active(X1),X2,X3) -> U101(X1,X2,X3)
  U101(X1,active(X2),X3) -> U101(X1,X2,X3)
  U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
  U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
  U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
  U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
  U102(active(X1),X2,X3) -> U102(X1,X2,X3)
  U102(X1,active(X2),X3) -> U102(X1,X2,X3)
  U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
  isNatKind(mark(X)) -> isNatKind(X)
  isNatKind(active(X)) -> isNatKind(X)
  U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
  U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
  U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
  U103(active(X1),X2,X3) -> U103(X1,X2,X3)
  U103(X1,active(X2),X3) -> U103(X1,X2,X3)
  U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
  isNatIListKind(mark(X)) -> isNatIListKind(X)
  isNatIListKind(active(X)) -> isNatIListKind(X)
  U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
  U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
  U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
  U104(active(X1),X2,X3) -> U104(X1,X2,X3)
  U104(X1,active(X2),X3) -> U104(X1,X2,X3)
  U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
  U105(mark(X1),X2) -> U105(X1,X2)
  U105(X1,mark(X2)) -> U105(X1,X2)
  U105(active(X1),X2) -> U105(X1,X2)
  U105(X1,active(X2)) -> U105(X1,X2)
  isNat(mark(X)) -> isNat(X)
  isNat(active(X)) -> isNat(X)
  U106(mark(X)) -> U106(X)
  U106(active(X)) -> U106(X)
  isNatIList(mark(X)) -> isNatIList(X)
  isNatIList(active(X)) -> isNatIList(X)
  U11(mark(X1),X2) -> U11(X1,X2)
  U11(X1,mark(X2)) -> U11(X1,X2)
  U11(active(X1),X2) -> U11(X1,X2)
  U11(X1,active(X2)) -> U11(X1,X2)
  U12(mark(X1),X2) -> U12(X1,X2)
  U12(X1,mark(X2)) -> U12(X1,X2)
  U12(active(X1),X2) -> U12(X1,X2)
  U12(X1,active(X2)) -> U12(X1,X2)
  U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
  U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
  U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
  U111(active(X1),X2,X3) -> U111(X1,X2,X3)
  U111(X1,active(X2),X3) -> U111(X1,X2,X3)
  U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
  U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
  U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
  U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
  U112(active(X1),X2,X3) -> U112(X1,X2,X3)
  U112(X1,active(X2),X3) -> U112(X1,X2,X3)
  U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
  U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
  U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
  U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
  U113(active(X1),X2,X3) -> U113(X1,X2,X3)
  U113(X1,active(X2),X3) -> U113(X1,X2,X3)
  U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
  U114(mark(X1),X2) -> U114(X1,X2)
  U114(X1,mark(X2)) -> U114(X1,X2)
  U114(active(X1),X2) -> U114(X1,X2)
  U114(X1,active(X2)) -> U114(X1,X2)
  s(mark(X)) -> s(X)
  s(active(X)) -> s(X)
  length(mark(X)) -> length(X)
  length(active(X)) -> length(X)
  U13(mark(X)) -> U13(X)
  U13(active(X)) -> U13(X)
  isNatList(mark(X)) -> isNatList(X)
  isNatList(active(X)) -> isNatList(X)
  U121(mark(X1),X2) -> U121(X1,X2)
  U121(X1,mark(X2)) -> U121(X1,X2)
  U121(active(X1),X2) -> U121(X1,X2)
  U121(X1,active(X2)) -> U121(X1,X2)
  U122(mark(X)) -> U122(X)
  U122(active(X)) -> U122(X)
  U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
  U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
  U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
  U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
  U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
  U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
  U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
  U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
  U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
  U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
  U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
  U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
  U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
  U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
  U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
  U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
  U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
  U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
  U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
  U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
  U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
  U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
  U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
  U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
  U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
  U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
  U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
  U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
  U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
  U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
  U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
  U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
  U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
  U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
  U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
  U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
  U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
  U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
  U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
  U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
  U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
  U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
  U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
  U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
  U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
  U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
  U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
  U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
  take(mark(X1),X2) -> take(X1,X2)
  take(X1,mark(X2)) -> take(X1,X2)
  take(active(X1),X2) -> take(X1,X2)
  take(X1,active(X2)) -> take(X1,X2)
  U21(mark(X1),X2) -> U21(X1,X2)
  U21(X1,mark(X2)) -> U21(X1,X2)
  U21(active(X1),X2) -> U21(X1,X2)
  U21(X1,active(X2)) -> U21(X1,X2)
  U22(mark(X1),X2) -> U22(X1,X2)
  U22(X1,mark(X2)) -> U22(X1,X2)
  U22(active(X1),X2) -> U22(X1,X2)
  U22(X1,active(X2)) -> U22(X1,X2)
  U23(mark(X)) -> U23(X)
  U23(active(X)) -> U23(X)
  U31(mark(X1),X2) -> U31(X1,X2)
  U31(X1,mark(X2)) -> U31(X1,X2)
  U31(active(X1),X2) -> U31(X1,X2)
  U31(X1,active(X2)) -> U31(X1,X2)
  U32(mark(X1),X2) -> U32(X1,X2)
  U32(X1,mark(X2)) -> U32(X1,X2)
  U32(active(X1),X2) -> U32(X1,X2)
  U32(X1,active(X2)) -> U32(X1,X2)
  U33(mark(X)) -> U33(X)
  U33(active(X)) -> U33(X)
  U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
  U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
  U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
  U41(active(X1),X2,X3) -> U41(X1,X2,X3)
  U41(X1,active(X2),X3) -> U41(X1,X2,X3)
  U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
  U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
  U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
  U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
  U42(active(X1),X2,X3) -> U42(X1,X2,X3)
  U42(X1,active(X2),X3) -> U42(X1,X2,X3)
  U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
  U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
  U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
  U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
  U43(active(X1),X2,X3) -> U43(X1,X2,X3)
  U43(X1,active(X2),X3) -> U43(X1,X2,X3)
  U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
  U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
  U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
  U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
  U44(active(X1),X2,X3) -> U44(X1,X2,X3)
  U44(X1,active(X2),X3) -> U44(X1,X2,X3)
  U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
  U45(mark(X1),X2) -> U45(X1,X2)
  U45(X1,mark(X2)) -> U45(X1,X2)
  U45(active(X1),X2) -> U45(X1,X2)
  U45(X1,active(X2)) -> U45(X1,X2)
  U46(mark(X)) -> U46(X)
  U46(active(X)) -> U46(X)
  U51(mark(X1),X2) -> U51(X1,X2)
  U51(X1,mark(X2)) -> U51(X1,X2)
  U51(active(X1),X2) -> U51(X1,X2)
  U51(X1,active(X2)) -> U51(X1,X2)
  U52(mark(X)) -> U52(X)
  U52(active(X)) -> U52(X)
  U61(mark(X1),X2) -> U61(X1,X2)
  U61(X1,mark(X2)) -> U61(X1,X2)
  U61(active(X1),X2) -> U61(X1,X2)
  U61(X1,active(X2)) -> U61(X1,X2)
  U62(mark(X)) -> U62(X)
  U62(active(X)) -> U62(X)
  U71(mark(X)) -> U71(X)
  U71(active(X)) -> U71(X)
  U81(mark(X)) -> U81(X)
  U81(active(X)) -> U81(X)
  U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
  U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
  U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
  U91(active(X1),X2,X3) -> U91(X1,X2,X3)
  U91(X1,active(X2),X3) -> U91(X1,X2,X3)
  U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
  U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
  U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
  U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
  U92(active(X1),X2,X3) -> U92(X1,X2,X3)
  U92(X1,active(X2),X3) -> U92(X1,X2,X3)
  U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
  U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
  U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
  U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
  U93(active(X1),X2,X3) -> U93(X1,X2,X3)
  U93(X1,active(X2),X3) -> U93(X1,X2,X3)
  U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
  U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
  U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
  U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
  U94(active(X1),X2,X3) -> U94(X1,X2,X3)
  U94(X1,active(X2),X3) -> U94(X1,X2,X3)
  U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
  U95(mark(X1),X2) -> U95(X1,X2)
  U95(X1,mark(X2)) -> U95(X1,X2)
  U95(active(X1),X2) -> U95(X1,X2)
  U95(X1,active(X2)) -> U95(X1,X2)
  U96(mark(X)) -> U96(X)
  U96(active(X)) -> U96(X)

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: active#(zeros()) -> mark#(cons(|0|(),zeros()))
p2: active#(zeros()) -> cons#(|0|(),zeros())
p3: active#(U101(tt(),V1,V2)) -> mark#(U102(isNatKind(V1),V1,V2))
p4: active#(U101(tt(),V1,V2)) -> U102#(isNatKind(V1),V1,V2)
p5: active#(U101(tt(),V1,V2)) -> isNatKind#(V1)
p6: active#(U102(tt(),V1,V2)) -> mark#(U103(isNatIListKind(V2),V1,V2))
p7: active#(U102(tt(),V1,V2)) -> U103#(isNatIListKind(V2),V1,V2)
p8: active#(U102(tt(),V1,V2)) -> isNatIListKind#(V2)
p9: active#(U103(tt(),V1,V2)) -> mark#(U104(isNatIListKind(V2),V1,V2))
p10: active#(U103(tt(),V1,V2)) -> U104#(isNatIListKind(V2),V1,V2)
p11: active#(U103(tt(),V1,V2)) -> isNatIListKind#(V2)
p12: active#(U104(tt(),V1,V2)) -> mark#(U105(isNat(V1),V2))
p13: active#(U104(tt(),V1,V2)) -> U105#(isNat(V1),V2)
p14: active#(U104(tt(),V1,V2)) -> isNat#(V1)
p15: active#(U105(tt(),V2)) -> mark#(U106(isNatIList(V2)))
p16: active#(U105(tt(),V2)) -> U106#(isNatIList(V2))
p17: active#(U105(tt(),V2)) -> isNatIList#(V2)
p18: active#(U106(tt())) -> mark#(tt())
p19: active#(U11(tt(),V1)) -> mark#(U12(isNatIListKind(V1),V1))
p20: active#(U11(tt(),V1)) -> U12#(isNatIListKind(V1),V1)
p21: active#(U11(tt(),V1)) -> isNatIListKind#(V1)
p22: active#(U111(tt(),L,N)) -> mark#(U112(isNatIListKind(L),L,N))
p23: active#(U111(tt(),L,N)) -> U112#(isNatIListKind(L),L,N)
p24: active#(U111(tt(),L,N)) -> isNatIListKind#(L)
p25: active#(U112(tt(),L,N)) -> mark#(U113(isNat(N),L,N))
p26: active#(U112(tt(),L,N)) -> U113#(isNat(N),L,N)
p27: active#(U112(tt(),L,N)) -> isNat#(N)
p28: active#(U113(tt(),L,N)) -> mark#(U114(isNatKind(N),L))
p29: active#(U113(tt(),L,N)) -> U114#(isNatKind(N),L)
p30: active#(U113(tt(),L,N)) -> isNatKind#(N)
p31: active#(U114(tt(),L)) -> mark#(s(length(L)))
p32: active#(U114(tt(),L)) -> s#(length(L))
p33: active#(U114(tt(),L)) -> length#(L)
p34: active#(U12(tt(),V1)) -> mark#(U13(isNatList(V1)))
p35: active#(U12(tt(),V1)) -> U13#(isNatList(V1))
p36: active#(U12(tt(),V1)) -> isNatList#(V1)
p37: active#(U121(tt(),IL)) -> mark#(U122(isNatIListKind(IL)))
p38: active#(U121(tt(),IL)) -> U122#(isNatIListKind(IL))
p39: active#(U121(tt(),IL)) -> isNatIListKind#(IL)
p40: active#(U122(tt())) -> mark#(nil())
p41: active#(U13(tt())) -> mark#(tt())
p42: active#(U131(tt(),IL,M,N)) -> mark#(U132(isNatIListKind(IL),IL,M,N))
p43: active#(U131(tt(),IL,M,N)) -> U132#(isNatIListKind(IL),IL,M,N)
p44: active#(U131(tt(),IL,M,N)) -> isNatIListKind#(IL)
p45: active#(U132(tt(),IL,M,N)) -> mark#(U133(isNat(M),IL,M,N))
p46: active#(U132(tt(),IL,M,N)) -> U133#(isNat(M),IL,M,N)
p47: active#(U132(tt(),IL,M,N)) -> isNat#(M)
p48: active#(U133(tt(),IL,M,N)) -> mark#(U134(isNatKind(M),IL,M,N))
p49: active#(U133(tt(),IL,M,N)) -> U134#(isNatKind(M),IL,M,N)
p50: active#(U133(tt(),IL,M,N)) -> isNatKind#(M)
p51: active#(U134(tt(),IL,M,N)) -> mark#(U135(isNat(N),IL,M,N))
p52: active#(U134(tt(),IL,M,N)) -> U135#(isNat(N),IL,M,N)
p53: active#(U134(tt(),IL,M,N)) -> isNat#(N)
p54: active#(U135(tt(),IL,M,N)) -> mark#(U136(isNatKind(N),IL,M,N))
p55: active#(U135(tt(),IL,M,N)) -> U136#(isNatKind(N),IL,M,N)
p56: active#(U135(tt(),IL,M,N)) -> isNatKind#(N)
p57: active#(U136(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL)))
p58: active#(U136(tt(),IL,M,N)) -> cons#(N,take(M,IL))
p59: active#(U136(tt(),IL,M,N)) -> take#(M,IL)
p60: active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1))
p61: active#(U21(tt(),V1)) -> U22#(isNatKind(V1),V1)
p62: active#(U21(tt(),V1)) -> isNatKind#(V1)
p63: active#(U22(tt(),V1)) -> mark#(U23(isNat(V1)))
p64: active#(U22(tt(),V1)) -> U23#(isNat(V1))
p65: active#(U22(tt(),V1)) -> isNat#(V1)
p66: active#(U23(tt())) -> mark#(tt())
p67: active#(U31(tt(),V)) -> mark#(U32(isNatIListKind(V),V))
p68: active#(U31(tt(),V)) -> U32#(isNatIListKind(V),V)
p69: active#(U31(tt(),V)) -> isNatIListKind#(V)
p70: active#(U32(tt(),V)) -> mark#(U33(isNatList(V)))
p71: active#(U32(tt(),V)) -> U33#(isNatList(V))
p72: active#(U32(tt(),V)) -> isNatList#(V)
p73: active#(U33(tt())) -> mark#(tt())
p74: active#(U41(tt(),V1,V2)) -> mark#(U42(isNatKind(V1),V1,V2))
p75: active#(U41(tt(),V1,V2)) -> U42#(isNatKind(V1),V1,V2)
p76: active#(U41(tt(),V1,V2)) -> isNatKind#(V1)
p77: active#(U42(tt(),V1,V2)) -> mark#(U43(isNatIListKind(V2),V1,V2))
p78: active#(U42(tt(),V1,V2)) -> U43#(isNatIListKind(V2),V1,V2)
p79: active#(U42(tt(),V1,V2)) -> isNatIListKind#(V2)
p80: active#(U43(tt(),V1,V2)) -> mark#(U44(isNatIListKind(V2),V1,V2))
p81: active#(U43(tt(),V1,V2)) -> U44#(isNatIListKind(V2),V1,V2)
p82: active#(U43(tt(),V1,V2)) -> isNatIListKind#(V2)
p83: active#(U44(tt(),V1,V2)) -> mark#(U45(isNat(V1),V2))
p84: active#(U44(tt(),V1,V2)) -> U45#(isNat(V1),V2)
p85: active#(U44(tt(),V1,V2)) -> isNat#(V1)
p86: active#(U45(tt(),V2)) -> mark#(U46(isNatIList(V2)))
p87: active#(U45(tt(),V2)) -> U46#(isNatIList(V2))
p88: active#(U45(tt(),V2)) -> isNatIList#(V2)
p89: active#(U46(tt())) -> mark#(tt())
p90: active#(U51(tt(),V2)) -> mark#(U52(isNatIListKind(V2)))
p91: active#(U51(tt(),V2)) -> U52#(isNatIListKind(V2))
p92: active#(U51(tt(),V2)) -> isNatIListKind#(V2)
p93: active#(U52(tt())) -> mark#(tt())
p94: active#(U61(tt(),V2)) -> mark#(U62(isNatIListKind(V2)))
p95: active#(U61(tt(),V2)) -> U62#(isNatIListKind(V2))
p96: active#(U61(tt(),V2)) -> isNatIListKind#(V2)
p97: active#(U62(tt())) -> mark#(tt())
p98: active#(U71(tt())) -> mark#(tt())
p99: active#(U81(tt())) -> mark#(tt())
p100: active#(U91(tt(),V1,V2)) -> mark#(U92(isNatKind(V1),V1,V2))
p101: active#(U91(tt(),V1,V2)) -> U92#(isNatKind(V1),V1,V2)
p102: active#(U91(tt(),V1,V2)) -> isNatKind#(V1)
p103: active#(U92(tt(),V1,V2)) -> mark#(U93(isNatIListKind(V2),V1,V2))
p104: active#(U92(tt(),V1,V2)) -> U93#(isNatIListKind(V2),V1,V2)
p105: active#(U92(tt(),V1,V2)) -> isNatIListKind#(V2)
p106: active#(U93(tt(),V1,V2)) -> mark#(U94(isNatIListKind(V2),V1,V2))
p107: active#(U93(tt(),V1,V2)) -> U94#(isNatIListKind(V2),V1,V2)
p108: active#(U93(tt(),V1,V2)) -> isNatIListKind#(V2)
p109: active#(U94(tt(),V1,V2)) -> mark#(U95(isNat(V1),V2))
p110: active#(U94(tt(),V1,V2)) -> U95#(isNat(V1),V2)
p111: active#(U94(tt(),V1,V2)) -> isNat#(V1)
p112: active#(U95(tt(),V2)) -> mark#(U96(isNatList(V2)))
p113: active#(U95(tt(),V2)) -> U96#(isNatList(V2))
p114: active#(U95(tt(),V2)) -> isNatList#(V2)
p115: active#(U96(tt())) -> mark#(tt())
p116: active#(isNat(|0|())) -> mark#(tt())
p117: active#(isNat(length(V1))) -> mark#(U11(isNatIListKind(V1),V1))
p118: active#(isNat(length(V1))) -> U11#(isNatIListKind(V1),V1)
p119: active#(isNat(length(V1))) -> isNatIListKind#(V1)
p120: active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1))
p121: active#(isNat(s(V1))) -> U21#(isNatKind(V1),V1)
p122: active#(isNat(s(V1))) -> isNatKind#(V1)
p123: active#(isNatIList(V)) -> mark#(U31(isNatIListKind(V),V))
p124: active#(isNatIList(V)) -> U31#(isNatIListKind(V),V)
p125: active#(isNatIList(V)) -> isNatIListKind#(V)
p126: active#(isNatIList(zeros())) -> mark#(tt())
p127: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNatKind(V1),V1,V2))
p128: active#(isNatIList(cons(V1,V2))) -> U41#(isNatKind(V1),V1,V2)
p129: active#(isNatIList(cons(V1,V2))) -> isNatKind#(V1)
p130: active#(isNatIListKind(nil())) -> mark#(tt())
p131: active#(isNatIListKind(zeros())) -> mark#(tt())
p132: active#(isNatIListKind(cons(V1,V2))) -> mark#(U51(isNatKind(V1),V2))
p133: active#(isNatIListKind(cons(V1,V2))) -> U51#(isNatKind(V1),V2)
p134: active#(isNatIListKind(cons(V1,V2))) -> isNatKind#(V1)
p135: active#(isNatIListKind(take(V1,V2))) -> mark#(U61(isNatKind(V1),V2))
p136: active#(isNatIListKind(take(V1,V2))) -> U61#(isNatKind(V1),V2)
p137: active#(isNatIListKind(take(V1,V2))) -> isNatKind#(V1)
p138: active#(isNatKind(|0|())) -> mark#(tt())
p139: active#(isNatKind(length(V1))) -> mark#(U71(isNatIListKind(V1)))
p140: active#(isNatKind(length(V1))) -> U71#(isNatIListKind(V1))
p141: active#(isNatKind(length(V1))) -> isNatIListKind#(V1)
p142: active#(isNatKind(s(V1))) -> mark#(U81(isNatKind(V1)))
p143: active#(isNatKind(s(V1))) -> U81#(isNatKind(V1))
p144: active#(isNatKind(s(V1))) -> isNatKind#(V1)
p145: active#(isNatList(nil())) -> mark#(tt())
p146: active#(isNatList(cons(V1,V2))) -> mark#(U91(isNatKind(V1),V1,V2))
p147: active#(isNatList(cons(V1,V2))) -> U91#(isNatKind(V1),V1,V2)
p148: active#(isNatList(cons(V1,V2))) -> isNatKind#(V1)
p149: active#(isNatList(take(V1,V2))) -> mark#(U101(isNatKind(V1),V1,V2))
p150: active#(isNatList(take(V1,V2))) -> U101#(isNatKind(V1),V1,V2)
p151: active#(isNatList(take(V1,V2))) -> isNatKind#(V1)
p152: active#(length(nil())) -> mark#(|0|())
p153: active#(length(cons(N,L))) -> mark#(U111(isNatList(L),L,N))
p154: active#(length(cons(N,L))) -> U111#(isNatList(L),L,N)
p155: active#(length(cons(N,L))) -> isNatList#(L)
p156: active#(take(|0|(),IL)) -> mark#(U121(isNatIList(IL),IL))
p157: active#(take(|0|(),IL)) -> U121#(isNatIList(IL),IL)
p158: active#(take(|0|(),IL)) -> isNatIList#(IL)
p159: active#(take(s(M),cons(N,IL))) -> mark#(U131(isNatIList(IL),IL,M,N))
p160: active#(take(s(M),cons(N,IL))) -> U131#(isNatIList(IL),IL,M,N)
p161: active#(take(s(M),cons(N,IL))) -> isNatIList#(IL)
p162: mark#(zeros()) -> active#(zeros())
p163: mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2))
p164: mark#(cons(X1,X2)) -> cons#(mark(X1),X2)
p165: mark#(cons(X1,X2)) -> mark#(X1)
p166: mark#(|0|()) -> active#(|0|())
p167: mark#(U101(X1,X2,X3)) -> active#(U101(mark(X1),X2,X3))
p168: mark#(U101(X1,X2,X3)) -> U101#(mark(X1),X2,X3)
p169: mark#(U101(X1,X2,X3)) -> mark#(X1)
p170: mark#(tt()) -> active#(tt())
p171: mark#(U102(X1,X2,X3)) -> active#(U102(mark(X1),X2,X3))
p172: mark#(U102(X1,X2,X3)) -> U102#(mark(X1),X2,X3)
p173: mark#(U102(X1,X2,X3)) -> mark#(X1)
p174: mark#(isNatKind(X)) -> active#(isNatKind(X))
p175: mark#(U103(X1,X2,X3)) -> active#(U103(mark(X1),X2,X3))
p176: mark#(U103(X1,X2,X3)) -> U103#(mark(X1),X2,X3)
p177: mark#(U103(X1,X2,X3)) -> mark#(X1)
p178: mark#(isNatIListKind(X)) -> active#(isNatIListKind(X))
p179: mark#(U104(X1,X2,X3)) -> active#(U104(mark(X1),X2,X3))
p180: mark#(U104(X1,X2,X3)) -> U104#(mark(X1),X2,X3)
p181: mark#(U104(X1,X2,X3)) -> mark#(X1)
p182: mark#(U105(X1,X2)) -> active#(U105(mark(X1),X2))
p183: mark#(U105(X1,X2)) -> U105#(mark(X1),X2)
p184: mark#(U105(X1,X2)) -> mark#(X1)
p185: mark#(isNat(X)) -> active#(isNat(X))
p186: mark#(U106(X)) -> active#(U106(mark(X)))
p187: mark#(U106(X)) -> U106#(mark(X))
p188: mark#(U106(X)) -> mark#(X)
p189: mark#(isNatIList(X)) -> active#(isNatIList(X))
p190: mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2))
p191: mark#(U11(X1,X2)) -> U11#(mark(X1),X2)
p192: mark#(U11(X1,X2)) -> mark#(X1)
p193: mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2))
p194: mark#(U12(X1,X2)) -> U12#(mark(X1),X2)
p195: mark#(U12(X1,X2)) -> mark#(X1)
p196: mark#(U111(X1,X2,X3)) -> active#(U111(mark(X1),X2,X3))
p197: mark#(U111(X1,X2,X3)) -> U111#(mark(X1),X2,X3)
p198: mark#(U111(X1,X2,X3)) -> mark#(X1)
p199: mark#(U112(X1,X2,X3)) -> active#(U112(mark(X1),X2,X3))
p200: mark#(U112(X1,X2,X3)) -> U112#(mark(X1),X2,X3)
p201: mark#(U112(X1,X2,X3)) -> mark#(X1)
p202: mark#(U113(X1,X2,X3)) -> active#(U113(mark(X1),X2,X3))
p203: mark#(U113(X1,X2,X3)) -> U113#(mark(X1),X2,X3)
p204: mark#(U113(X1,X2,X3)) -> mark#(X1)
p205: mark#(U114(X1,X2)) -> active#(U114(mark(X1),X2))
p206: mark#(U114(X1,X2)) -> U114#(mark(X1),X2)
p207: mark#(U114(X1,X2)) -> mark#(X1)
p208: mark#(s(X)) -> active#(s(mark(X)))
p209: mark#(s(X)) -> s#(mark(X))
p210: mark#(s(X)) -> mark#(X)
p211: mark#(length(X)) -> active#(length(mark(X)))
p212: mark#(length(X)) -> length#(mark(X))
p213: mark#(length(X)) -> mark#(X)
p214: mark#(U13(X)) -> active#(U13(mark(X)))
p215: mark#(U13(X)) -> U13#(mark(X))
p216: mark#(U13(X)) -> mark#(X)
p217: mark#(isNatList(X)) -> active#(isNatList(X))
p218: mark#(U121(X1,X2)) -> active#(U121(mark(X1),X2))
p219: mark#(U121(X1,X2)) -> U121#(mark(X1),X2)
p220: mark#(U121(X1,X2)) -> mark#(X1)
p221: mark#(U122(X)) -> active#(U122(mark(X)))
p222: mark#(U122(X)) -> U122#(mark(X))
p223: mark#(U122(X)) -> mark#(X)
p224: mark#(nil()) -> active#(nil())
p225: mark#(U131(X1,X2,X3,X4)) -> active#(U131(mark(X1),X2,X3,X4))
p226: mark#(U131(X1,X2,X3,X4)) -> U131#(mark(X1),X2,X3,X4)
p227: mark#(U131(X1,X2,X3,X4)) -> mark#(X1)
p228: mark#(U132(X1,X2,X3,X4)) -> active#(U132(mark(X1),X2,X3,X4))
p229: mark#(U132(X1,X2,X3,X4)) -> U132#(mark(X1),X2,X3,X4)
p230: mark#(U132(X1,X2,X3,X4)) -> mark#(X1)
p231: mark#(U133(X1,X2,X3,X4)) -> active#(U133(mark(X1),X2,X3,X4))
p232: mark#(U133(X1,X2,X3,X4)) -> U133#(mark(X1),X2,X3,X4)
p233: mark#(U133(X1,X2,X3,X4)) -> mark#(X1)
p234: mark#(U134(X1,X2,X3,X4)) -> active#(U134(mark(X1),X2,X3,X4))
p235: mark#(U134(X1,X2,X3,X4)) -> U134#(mark(X1),X2,X3,X4)
p236: mark#(U134(X1,X2,X3,X4)) -> mark#(X1)
p237: mark#(U135(X1,X2,X3,X4)) -> active#(U135(mark(X1),X2,X3,X4))
p238: mark#(U135(X1,X2,X3,X4)) -> U135#(mark(X1),X2,X3,X4)
p239: mark#(U135(X1,X2,X3,X4)) -> mark#(X1)
p240: mark#(U136(X1,X2,X3,X4)) -> active#(U136(mark(X1),X2,X3,X4))
p241: mark#(U136(X1,X2,X3,X4)) -> U136#(mark(X1),X2,X3,X4)
p242: mark#(U136(X1,X2,X3,X4)) -> mark#(X1)
p243: mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2)))
p244: mark#(take(X1,X2)) -> take#(mark(X1),mark(X2))
p245: mark#(take(X1,X2)) -> mark#(X1)
p246: mark#(take(X1,X2)) -> mark#(X2)
p247: mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2))
p248: mark#(U21(X1,X2)) -> U21#(mark(X1),X2)
p249: mark#(U21(X1,X2)) -> mark#(X1)
p250: mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2))
p251: mark#(U22(X1,X2)) -> U22#(mark(X1),X2)
p252: mark#(U22(X1,X2)) -> mark#(X1)
p253: mark#(U23(X)) -> active#(U23(mark(X)))
p254: mark#(U23(X)) -> U23#(mark(X))
p255: mark#(U23(X)) -> mark#(X)
p256: mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2))
p257: mark#(U31(X1,X2)) -> U31#(mark(X1),X2)
p258: mark#(U31(X1,X2)) -> mark#(X1)
p259: mark#(U32(X1,X2)) -> active#(U32(mark(X1),X2))
p260: mark#(U32(X1,X2)) -> U32#(mark(X1),X2)
p261: mark#(U32(X1,X2)) -> mark#(X1)
p262: mark#(U33(X)) -> active#(U33(mark(X)))
p263: mark#(U33(X)) -> U33#(mark(X))
p264: mark#(U33(X)) -> mark#(X)
p265: mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3))
p266: mark#(U41(X1,X2,X3)) -> U41#(mark(X1),X2,X3)
p267: mark#(U41(X1,X2,X3)) -> mark#(X1)
p268: mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3))
p269: mark#(U42(X1,X2,X3)) -> U42#(mark(X1),X2,X3)
p270: mark#(U42(X1,X2,X3)) -> mark#(X1)
p271: mark#(U43(X1,X2,X3)) -> active#(U43(mark(X1),X2,X3))
p272: mark#(U43(X1,X2,X3)) -> U43#(mark(X1),X2,X3)
p273: mark#(U43(X1,X2,X3)) -> mark#(X1)
p274: mark#(U44(X1,X2,X3)) -> active#(U44(mark(X1),X2,X3))
p275: mark#(U44(X1,X2,X3)) -> U44#(mark(X1),X2,X3)
p276: mark#(U44(X1,X2,X3)) -> mark#(X1)
p277: mark#(U45(X1,X2)) -> active#(U45(mark(X1),X2))
p278: mark#(U45(X1,X2)) -> U45#(mark(X1),X2)
p279: mark#(U45(X1,X2)) -> mark#(X1)
p280: mark#(U46(X)) -> active#(U46(mark(X)))
p281: mark#(U46(X)) -> U46#(mark(X))
p282: mark#(U46(X)) -> mark#(X)
p283: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p284: mark#(U51(X1,X2)) -> U51#(mark(X1),X2)
p285: mark#(U51(X1,X2)) -> mark#(X1)
p286: mark#(U52(X)) -> active#(U52(mark(X)))
p287: mark#(U52(X)) -> U52#(mark(X))
p288: mark#(U52(X)) -> mark#(X)
p289: mark#(U61(X1,X2)) -> active#(U61(mark(X1),X2))
p290: mark#(U61(X1,X2)) -> U61#(mark(X1),X2)
p291: mark#(U61(X1,X2)) -> mark#(X1)
p292: mark#(U62(X)) -> active#(U62(mark(X)))
p293: mark#(U62(X)) -> U62#(mark(X))
p294: mark#(U62(X)) -> mark#(X)
p295: mark#(U71(X)) -> active#(U71(mark(X)))
p296: mark#(U71(X)) -> U71#(mark(X))
p297: mark#(U71(X)) -> mark#(X)
p298: mark#(U81(X)) -> active#(U81(mark(X)))
p299: mark#(U81(X)) -> U81#(mark(X))
p300: mark#(U81(X)) -> mark#(X)
p301: mark#(U91(X1,X2,X3)) -> active#(U91(mark(X1),X2,X3))
p302: mark#(U91(X1,X2,X3)) -> U91#(mark(X1),X2,X3)
p303: mark#(U91(X1,X2,X3)) -> mark#(X1)
p304: mark#(U92(X1,X2,X3)) -> active#(U92(mark(X1),X2,X3))
p305: mark#(U92(X1,X2,X3)) -> U92#(mark(X1),X2,X3)
p306: mark#(U92(X1,X2,X3)) -> mark#(X1)
p307: mark#(U93(X1,X2,X3)) -> active#(U93(mark(X1),X2,X3))
p308: mark#(U93(X1,X2,X3)) -> U93#(mark(X1),X2,X3)
p309: mark#(U93(X1,X2,X3)) -> mark#(X1)
p310: mark#(U94(X1,X2,X3)) -> active#(U94(mark(X1),X2,X3))
p311: mark#(U94(X1,X2,X3)) -> U94#(mark(X1),X2,X3)
p312: mark#(U94(X1,X2,X3)) -> mark#(X1)
p313: mark#(U95(X1,X2)) -> active#(U95(mark(X1),X2))
p314: mark#(U95(X1,X2)) -> U95#(mark(X1),X2)
p315: mark#(U95(X1,X2)) -> mark#(X1)
p316: mark#(U96(X)) -> active#(U96(mark(X)))
p317: mark#(U96(X)) -> U96#(mark(X))
p318: mark#(U96(X)) -> mark#(X)
p319: cons#(mark(X1),X2) -> cons#(X1,X2)
p320: cons#(X1,mark(X2)) -> cons#(X1,X2)
p321: cons#(active(X1),X2) -> cons#(X1,X2)
p322: cons#(X1,active(X2)) -> cons#(X1,X2)
p323: U101#(mark(X1),X2,X3) -> U101#(X1,X2,X3)
p324: U101#(X1,mark(X2),X3) -> U101#(X1,X2,X3)
p325: U101#(X1,X2,mark(X3)) -> U101#(X1,X2,X3)
p326: U101#(active(X1),X2,X3) -> U101#(X1,X2,X3)
p327: U101#(X1,active(X2),X3) -> U101#(X1,X2,X3)
p328: U101#(X1,X2,active(X3)) -> U101#(X1,X2,X3)
p329: U102#(mark(X1),X2,X3) -> U102#(X1,X2,X3)
p330: U102#(X1,mark(X2),X3) -> U102#(X1,X2,X3)
p331: U102#(X1,X2,mark(X3)) -> U102#(X1,X2,X3)
p332: U102#(active(X1),X2,X3) -> U102#(X1,X2,X3)
p333: U102#(X1,active(X2),X3) -> U102#(X1,X2,X3)
p334: U102#(X1,X2,active(X3)) -> U102#(X1,X2,X3)
p335: isNatKind#(mark(X)) -> isNatKind#(X)
p336: isNatKind#(active(X)) -> isNatKind#(X)
p337: U103#(mark(X1),X2,X3) -> U103#(X1,X2,X3)
p338: U103#(X1,mark(X2),X3) -> U103#(X1,X2,X3)
p339: U103#(X1,X2,mark(X3)) -> U103#(X1,X2,X3)
p340: U103#(active(X1),X2,X3) -> U103#(X1,X2,X3)
p341: U103#(X1,active(X2),X3) -> U103#(X1,X2,X3)
p342: U103#(X1,X2,active(X3)) -> U103#(X1,X2,X3)
p343: isNatIListKind#(mark(X)) -> isNatIListKind#(X)
p344: isNatIListKind#(active(X)) -> isNatIListKind#(X)
p345: U104#(mark(X1),X2,X3) -> U104#(X1,X2,X3)
p346: U104#(X1,mark(X2),X3) -> U104#(X1,X2,X3)
p347: U104#(X1,X2,mark(X3)) -> U104#(X1,X2,X3)
p348: U104#(active(X1),X2,X3) -> U104#(X1,X2,X3)
p349: U104#(X1,active(X2),X3) -> U104#(X1,X2,X3)
p350: U104#(X1,X2,active(X3)) -> U104#(X1,X2,X3)
p351: U105#(mark(X1),X2) -> U105#(X1,X2)
p352: U105#(X1,mark(X2)) -> U105#(X1,X2)
p353: U105#(active(X1),X2) -> U105#(X1,X2)
p354: U105#(X1,active(X2)) -> U105#(X1,X2)
p355: isNat#(mark(X)) -> isNat#(X)
p356: isNat#(active(X)) -> isNat#(X)
p357: U106#(mark(X)) -> U106#(X)
p358: U106#(active(X)) -> U106#(X)
p359: isNatIList#(mark(X)) -> isNatIList#(X)
p360: isNatIList#(active(X)) -> isNatIList#(X)
p361: U11#(mark(X1),X2) -> U11#(X1,X2)
p362: U11#(X1,mark(X2)) -> U11#(X1,X2)
p363: U11#(active(X1),X2) -> U11#(X1,X2)
p364: U11#(X1,active(X2)) -> U11#(X1,X2)
p365: U12#(mark(X1),X2) -> U12#(X1,X2)
p366: U12#(X1,mark(X2)) -> U12#(X1,X2)
p367: U12#(active(X1),X2) -> U12#(X1,X2)
p368: U12#(X1,active(X2)) -> U12#(X1,X2)
p369: U111#(mark(X1),X2,X3) -> U111#(X1,X2,X3)
p370: U111#(X1,mark(X2),X3) -> U111#(X1,X2,X3)
p371: U111#(X1,X2,mark(X3)) -> U111#(X1,X2,X3)
p372: U111#(active(X1),X2,X3) -> U111#(X1,X2,X3)
p373: U111#(X1,active(X2),X3) -> U111#(X1,X2,X3)
p374: U111#(X1,X2,active(X3)) -> U111#(X1,X2,X3)
p375: U112#(mark(X1),X2,X3) -> U112#(X1,X2,X3)
p376: U112#(X1,mark(X2),X3) -> U112#(X1,X2,X3)
p377: U112#(X1,X2,mark(X3)) -> U112#(X1,X2,X3)
p378: U112#(active(X1),X2,X3) -> U112#(X1,X2,X3)
p379: U112#(X1,active(X2),X3) -> U112#(X1,X2,X3)
p380: U112#(X1,X2,active(X3)) -> U112#(X1,X2,X3)
p381: U113#(mark(X1),X2,X3) -> U113#(X1,X2,X3)
p382: U113#(X1,mark(X2),X3) -> U113#(X1,X2,X3)
p383: U113#(X1,X2,mark(X3)) -> U113#(X1,X2,X3)
p384: U113#(active(X1),X2,X3) -> U113#(X1,X2,X3)
p385: U113#(X1,active(X2),X3) -> U113#(X1,X2,X3)
p386: U113#(X1,X2,active(X3)) -> U113#(X1,X2,X3)
p387: U114#(mark(X1),X2) -> U114#(X1,X2)
p388: U114#(X1,mark(X2)) -> U114#(X1,X2)
p389: U114#(active(X1),X2) -> U114#(X1,X2)
p390: U114#(X1,active(X2)) -> U114#(X1,X2)
p391: s#(mark(X)) -> s#(X)
p392: s#(active(X)) -> s#(X)
p393: length#(mark(X)) -> length#(X)
p394: length#(active(X)) -> length#(X)
p395: U13#(mark(X)) -> U13#(X)
p396: U13#(active(X)) -> U13#(X)
p397: isNatList#(mark(X)) -> isNatList#(X)
p398: isNatList#(active(X)) -> isNatList#(X)
p399: U121#(mark(X1),X2) -> U121#(X1,X2)
p400: U121#(X1,mark(X2)) -> U121#(X1,X2)
p401: U121#(active(X1),X2) -> U121#(X1,X2)
p402: U121#(X1,active(X2)) -> U121#(X1,X2)
p403: U122#(mark(X)) -> U122#(X)
p404: U122#(active(X)) -> U122#(X)
p405: U131#(mark(X1),X2,X3,X4) -> U131#(X1,X2,X3,X4)
p406: U131#(X1,mark(X2),X3,X4) -> U131#(X1,X2,X3,X4)
p407: U131#(X1,X2,mark(X3),X4) -> U131#(X1,X2,X3,X4)
p408: U131#(X1,X2,X3,mark(X4)) -> U131#(X1,X2,X3,X4)
p409: U131#(active(X1),X2,X3,X4) -> U131#(X1,X2,X3,X4)
p410: U131#(X1,active(X2),X3,X4) -> U131#(X1,X2,X3,X4)
p411: U131#(X1,X2,active(X3),X4) -> U131#(X1,X2,X3,X4)
p412: U131#(X1,X2,X3,active(X4)) -> U131#(X1,X2,X3,X4)
p413: U132#(mark(X1),X2,X3,X4) -> U132#(X1,X2,X3,X4)
p414: U132#(X1,mark(X2),X3,X4) -> U132#(X1,X2,X3,X4)
p415: U132#(X1,X2,mark(X3),X4) -> U132#(X1,X2,X3,X4)
p416: U132#(X1,X2,X3,mark(X4)) -> U132#(X1,X2,X3,X4)
p417: U132#(active(X1),X2,X3,X4) -> U132#(X1,X2,X3,X4)
p418: U132#(X1,active(X2),X3,X4) -> U132#(X1,X2,X3,X4)
p419: U132#(X1,X2,active(X3),X4) -> U132#(X1,X2,X3,X4)
p420: U132#(X1,X2,X3,active(X4)) -> U132#(X1,X2,X3,X4)
p421: U133#(mark(X1),X2,X3,X4) -> U133#(X1,X2,X3,X4)
p422: U133#(X1,mark(X2),X3,X4) -> U133#(X1,X2,X3,X4)
p423: U133#(X1,X2,mark(X3),X4) -> U133#(X1,X2,X3,X4)
p424: U133#(X1,X2,X3,mark(X4)) -> U133#(X1,X2,X3,X4)
p425: U133#(active(X1),X2,X3,X4) -> U133#(X1,X2,X3,X4)
p426: U133#(X1,active(X2),X3,X4) -> U133#(X1,X2,X3,X4)
p427: U133#(X1,X2,active(X3),X4) -> U133#(X1,X2,X3,X4)
p428: U133#(X1,X2,X3,active(X4)) -> U133#(X1,X2,X3,X4)
p429: U134#(mark(X1),X2,X3,X4) -> U134#(X1,X2,X3,X4)
p430: U134#(X1,mark(X2),X3,X4) -> U134#(X1,X2,X3,X4)
p431: U134#(X1,X2,mark(X3),X4) -> U134#(X1,X2,X3,X4)
p432: U134#(X1,X2,X3,mark(X4)) -> U134#(X1,X2,X3,X4)
p433: U134#(active(X1),X2,X3,X4) -> U134#(X1,X2,X3,X4)
p434: U134#(X1,active(X2),X3,X4) -> U134#(X1,X2,X3,X4)
p435: U134#(X1,X2,active(X3),X4) -> U134#(X1,X2,X3,X4)
p436: U134#(X1,X2,X3,active(X4)) -> U134#(X1,X2,X3,X4)
p437: U135#(mark(X1),X2,X3,X4) -> U135#(X1,X2,X3,X4)
p438: U135#(X1,mark(X2),X3,X4) -> U135#(X1,X2,X3,X4)
p439: U135#(X1,X2,mark(X3),X4) -> U135#(X1,X2,X3,X4)
p440: U135#(X1,X2,X3,mark(X4)) -> U135#(X1,X2,X3,X4)
p441: U135#(active(X1),X2,X3,X4) -> U135#(X1,X2,X3,X4)
p442: U135#(X1,active(X2),X3,X4) -> U135#(X1,X2,X3,X4)
p443: U135#(X1,X2,active(X3),X4) -> U135#(X1,X2,X3,X4)
p444: U135#(X1,X2,X3,active(X4)) -> U135#(X1,X2,X3,X4)
p445: U136#(mark(X1),X2,X3,X4) -> U136#(X1,X2,X3,X4)
p446: U136#(X1,mark(X2),X3,X4) -> U136#(X1,X2,X3,X4)
p447: U136#(X1,X2,mark(X3),X4) -> U136#(X1,X2,X3,X4)
p448: U136#(X1,X2,X3,mark(X4)) -> U136#(X1,X2,X3,X4)
p449: U136#(active(X1),X2,X3,X4) -> U136#(X1,X2,X3,X4)
p450: U136#(X1,active(X2),X3,X4) -> U136#(X1,X2,X3,X4)
p451: U136#(X1,X2,active(X3),X4) -> U136#(X1,X2,X3,X4)
p452: U136#(X1,X2,X3,active(X4)) -> U136#(X1,X2,X3,X4)
p453: take#(mark(X1),X2) -> take#(X1,X2)
p454: take#(X1,mark(X2)) -> take#(X1,X2)
p455: take#(active(X1),X2) -> take#(X1,X2)
p456: take#(X1,active(X2)) -> take#(X1,X2)
p457: U21#(mark(X1),X2) -> U21#(X1,X2)
p458: U21#(X1,mark(X2)) -> U21#(X1,X2)
p459: U21#(active(X1),X2) -> U21#(X1,X2)
p460: U21#(X1,active(X2)) -> U21#(X1,X2)
p461: U22#(mark(X1),X2) -> U22#(X1,X2)
p462: U22#(X1,mark(X2)) -> U22#(X1,X2)
p463: U22#(active(X1),X2) -> U22#(X1,X2)
p464: U22#(X1,active(X2)) -> U22#(X1,X2)
p465: U23#(mark(X)) -> U23#(X)
p466: U23#(active(X)) -> U23#(X)
p467: U31#(mark(X1),X2) -> U31#(X1,X2)
p468: U31#(X1,mark(X2)) -> U31#(X1,X2)
p469: U31#(active(X1),X2) -> U31#(X1,X2)
p470: U31#(X1,active(X2)) -> U31#(X1,X2)
p471: U32#(mark(X1),X2) -> U32#(X1,X2)
p472: U32#(X1,mark(X2)) -> U32#(X1,X2)
p473: U32#(active(X1),X2) -> U32#(X1,X2)
p474: U32#(X1,active(X2)) -> U32#(X1,X2)
p475: U33#(mark(X)) -> U33#(X)
p476: U33#(active(X)) -> U33#(X)
p477: U41#(mark(X1),X2,X3) -> U41#(X1,X2,X3)
p478: U41#(X1,mark(X2),X3) -> U41#(X1,X2,X3)
p479: U41#(X1,X2,mark(X3)) -> U41#(X1,X2,X3)
p480: U41#(active(X1),X2,X3) -> U41#(X1,X2,X3)
p481: U41#(X1,active(X2),X3) -> U41#(X1,X2,X3)
p482: U41#(X1,X2,active(X3)) -> U41#(X1,X2,X3)
p483: U42#(mark(X1),X2,X3) -> U42#(X1,X2,X3)
p484: U42#(X1,mark(X2),X3) -> U42#(X1,X2,X3)
p485: U42#(X1,X2,mark(X3)) -> U42#(X1,X2,X3)
p486: U42#(active(X1),X2,X3) -> U42#(X1,X2,X3)
p487: U42#(X1,active(X2),X3) -> U42#(X1,X2,X3)
p488: U42#(X1,X2,active(X3)) -> U42#(X1,X2,X3)
p489: U43#(mark(X1),X2,X3) -> U43#(X1,X2,X3)
p490: U43#(X1,mark(X2),X3) -> U43#(X1,X2,X3)
p491: U43#(X1,X2,mark(X3)) -> U43#(X1,X2,X3)
p492: U43#(active(X1),X2,X3) -> U43#(X1,X2,X3)
p493: U43#(X1,active(X2),X3) -> U43#(X1,X2,X3)
p494: U43#(X1,X2,active(X3)) -> U43#(X1,X2,X3)
p495: U44#(mark(X1),X2,X3) -> U44#(X1,X2,X3)
p496: U44#(X1,mark(X2),X3) -> U44#(X1,X2,X3)
p497: U44#(X1,X2,mark(X3)) -> U44#(X1,X2,X3)
p498: U44#(active(X1),X2,X3) -> U44#(X1,X2,X3)
p499: U44#(X1,active(X2),X3) -> U44#(X1,X2,X3)
p500: U44#(X1,X2,active(X3)) -> U44#(X1,X2,X3)
p501: U45#(mark(X1),X2) -> U45#(X1,X2)
p502: U45#(X1,mark(X2)) -> U45#(X1,X2)
p503: U45#(active(X1),X2) -> U45#(X1,X2)
p504: U45#(X1,active(X2)) -> U45#(X1,X2)
p505: U46#(mark(X)) -> U46#(X)
p506: U46#(active(X)) -> U46#(X)
p507: U51#(mark(X1),X2) -> U51#(X1,X2)
p508: U51#(X1,mark(X2)) -> U51#(X1,X2)
p509: U51#(active(X1),X2) -> U51#(X1,X2)
p510: U51#(X1,active(X2)) -> U51#(X1,X2)
p511: U52#(mark(X)) -> U52#(X)
p512: U52#(active(X)) -> U52#(X)
p513: U61#(mark(X1),X2) -> U61#(X1,X2)
p514: U61#(X1,mark(X2)) -> U61#(X1,X2)
p515: U61#(active(X1),X2) -> U61#(X1,X2)
p516: U61#(X1,active(X2)) -> U61#(X1,X2)
p517: U62#(mark(X)) -> U62#(X)
p518: U62#(active(X)) -> U62#(X)
p519: U71#(mark(X)) -> U71#(X)
p520: U71#(active(X)) -> U71#(X)
p521: U81#(mark(X)) -> U81#(X)
p522: U81#(active(X)) -> U81#(X)
p523: U91#(mark(X1),X2,X3) -> U91#(X1,X2,X3)
p524: U91#(X1,mark(X2),X3) -> U91#(X1,X2,X3)
p525: U91#(X1,X2,mark(X3)) -> U91#(X1,X2,X3)
p526: U91#(active(X1),X2,X3) -> U91#(X1,X2,X3)
p527: U91#(X1,active(X2),X3) -> U91#(X1,X2,X3)
p528: U91#(X1,X2,active(X3)) -> U91#(X1,X2,X3)
p529: U92#(mark(X1),X2,X3) -> U92#(X1,X2,X3)
p530: U92#(X1,mark(X2),X3) -> U92#(X1,X2,X3)
p531: U92#(X1,X2,mark(X3)) -> U92#(X1,X2,X3)
p532: U92#(active(X1),X2,X3) -> U92#(X1,X2,X3)
p533: U92#(X1,active(X2),X3) -> U92#(X1,X2,X3)
p534: U92#(X1,X2,active(X3)) -> U92#(X1,X2,X3)
p535: U93#(mark(X1),X2,X3) -> U93#(X1,X2,X3)
p536: U93#(X1,mark(X2),X3) -> U93#(X1,X2,X3)
p537: U93#(X1,X2,mark(X3)) -> U93#(X1,X2,X3)
p538: U93#(active(X1),X2,X3) -> U93#(X1,X2,X3)
p539: U93#(X1,active(X2),X3) -> U93#(X1,X2,X3)
p540: U93#(X1,X2,active(X3)) -> U93#(X1,X2,X3)
p541: U94#(mark(X1),X2,X3) -> U94#(X1,X2,X3)
p542: U94#(X1,mark(X2),X3) -> U94#(X1,X2,X3)
p543: U94#(X1,X2,mark(X3)) -> U94#(X1,X2,X3)
p544: U94#(active(X1),X2,X3) -> U94#(X1,X2,X3)
p545: U94#(X1,active(X2),X3) -> U94#(X1,X2,X3)
p546: U94#(X1,X2,active(X3)) -> U94#(X1,X2,X3)
p547: U95#(mark(X1),X2) -> U95#(X1,X2)
p548: U95#(X1,mark(X2)) -> U95#(X1,X2)
p549: U95#(active(X1),X2) -> U95#(X1,X2)
p550: U95#(X1,active(X2)) -> U95#(X1,X2)
p551: U96#(mark(X)) -> U96#(X)
p552: U96#(active(X)) -> U96#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p3, p6, p9, p12, p15, p19, p22, p25, p28, p31, p34, p37, p42, p45, p48, p51, p54, p57, p60, p63, p67, p70, p74, p77, p80, p83, p86, p90, p94, p100, p103, p106, p109, p112, p117, p120, p123, p127, p132, p135, p139, p142, p146, p149, p153, p156, p159, p162, p163, p165, p167, p169, p171, p173, p174, p175, p177, p178, p179, p181, p182, p184, p185, p186, p188, p189, p190, p192, p193, p195, p196, p198, p199, p201, p202, p204, p205, p207, p208, p210, p211, p213, p214, p216, p217, p218, p220, p221, p223, p225, p227, p228, p230, p231, p233, p234, p236, p237, p239, p240, p242, p243, p245, p246, p247, p249, p250, p252, p253, p255, p256, p258, p259, p261, p262, p264, p265, p267, p268, p270, p271, p273, p274, p276, p277, p279, p280, p282, p283, p285, p286, p288, p289, p291, p292, p294, p295, p297, p298, p300, p301, p303, p304, p306, p307, p309, p310, p312, p313, p315, p316, p318}
  {p329, p330, p331, p332, p333, p334}
  {p335, p336}
  {p337, p338, p339, p340, p341, p342}
  {p343, p344}
  {p345, p346, p347, p348, p349, p350}
  {p351, p352, p353, p354}
  {p355, p356}
  {p357, p358}
  {p359, p360}
  {p365, p366, p367, p368}
  {p375, p376, p377, p378, p379, p380}
  {p381, p382, p383, p384, p385, p386}
  {p387, p388, p389, p390}
  {p391, p392}
  {p393, p394}
  {p395, p396}
  {p397, p398}
  {p403, p404}
  {p413, p414, p415, p416, p417, p418, p419, p420}
  {p421, p422, p423, p424, p425, p426, p427, p428}
  {p429, p430, p431, p432, p433, p434, p435, p436}
  {p437, p438, p439, p440, p441, p442, p443, p444}
  {p445, p446, p447, p448, p449, p450, p451, p452}
  {p319, p320, p321, p322}
  {p453, p454, p455, p456}
  {p461, p462, p463, p464}
  {p465, p466}
  {p471, p472, p473, p474}
  {p475, p476}
  {p483, p484, p485, p486, p487, p488}
  {p489, p490, p491, p492, p493, p494}
  {p495, p496, p497, p498, p499, p500}
  {p501, p502, p503, p504}
  {p505, p506}
  {p511, p512}
  {p517, p518}
  {p529, p530, p531, p532, p533, p534}
  {p535, p536, p537, p538, p539, p540}
  {p541, p542, p543, p544, p545, p546}
  {p547, p548, p549, p550}
  {p551, p552}
  {p361, p362, p363, p364}
  {p457, p458, p459, p460}
  {p467, p468, p469, p470}
  {p477, p478, p479, p480, p481, p482}
  {p507, p508, p509, p510}
  {p513, p514, p515, p516}
  {p519, p520}
  {p521, p522}
  {p523, p524, p525, p526, p527, p528}
  {p323, p324, p325, p326, p327, p328}
  {p369, p370, p371, p372, p373, p374}
  {p399, p400, p401, p402}
  {p405, p406, p407, p408, p409, p410, p411, p412}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: active#(zeros()) -> mark#(cons(|0|(),zeros()))
p2: mark#(cons(X1,X2)) -> mark#(X1)
p3: mark#(U96(X)) -> mark#(X)
p4: mark#(U96(X)) -> active#(U96(mark(X)))
p5: active#(take(s(M),cons(N,IL))) -> mark#(U131(isNatIList(IL),IL,M,N))
p6: mark#(U95(X1,X2)) -> mark#(X1)
p7: mark#(U95(X1,X2)) -> active#(U95(mark(X1),X2))
p8: active#(take(|0|(),IL)) -> mark#(U121(isNatIList(IL),IL))
p9: mark#(U94(X1,X2,X3)) -> mark#(X1)
p10: mark#(U94(X1,X2,X3)) -> active#(U94(mark(X1),X2,X3))
p11: active#(length(cons(N,L))) -> mark#(U111(isNatList(L),L,N))
p12: mark#(U93(X1,X2,X3)) -> mark#(X1)
p13: mark#(U93(X1,X2,X3)) -> active#(U93(mark(X1),X2,X3))
p14: active#(isNatList(take(V1,V2))) -> mark#(U101(isNatKind(V1),V1,V2))
p15: mark#(U92(X1,X2,X3)) -> mark#(X1)
p16: mark#(U92(X1,X2,X3)) -> active#(U92(mark(X1),X2,X3))
p17: active#(isNatList(cons(V1,V2))) -> mark#(U91(isNatKind(V1),V1,V2))
p18: mark#(U91(X1,X2,X3)) -> mark#(X1)
p19: mark#(U91(X1,X2,X3)) -> active#(U91(mark(X1),X2,X3))
p20: active#(isNatKind(s(V1))) -> mark#(U81(isNatKind(V1)))
p21: mark#(U81(X)) -> mark#(X)
p22: mark#(U81(X)) -> active#(U81(mark(X)))
p23: active#(isNatKind(length(V1))) -> mark#(U71(isNatIListKind(V1)))
p24: mark#(U71(X)) -> mark#(X)
p25: mark#(U71(X)) -> active#(U71(mark(X)))
p26: active#(isNatIListKind(take(V1,V2))) -> mark#(U61(isNatKind(V1),V2))
p27: mark#(U62(X)) -> mark#(X)
p28: mark#(U62(X)) -> active#(U62(mark(X)))
p29: active#(isNatIListKind(cons(V1,V2))) -> mark#(U51(isNatKind(V1),V2))
p30: mark#(U61(X1,X2)) -> mark#(X1)
p31: mark#(U61(X1,X2)) -> active#(U61(mark(X1),X2))
p32: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNatKind(V1),V1,V2))
p33: mark#(U52(X)) -> mark#(X)
p34: mark#(U52(X)) -> active#(U52(mark(X)))
p35: active#(isNatIList(V)) -> mark#(U31(isNatIListKind(V),V))
p36: mark#(U51(X1,X2)) -> mark#(X1)
p37: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p38: active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1))
p39: mark#(U46(X)) -> mark#(X)
p40: mark#(U46(X)) -> active#(U46(mark(X)))
p41: active#(isNat(length(V1))) -> mark#(U11(isNatIListKind(V1),V1))
p42: mark#(U45(X1,X2)) -> mark#(X1)
p43: mark#(U45(X1,X2)) -> active#(U45(mark(X1),X2))
p44: active#(U95(tt(),V2)) -> mark#(U96(isNatList(V2)))
p45: mark#(U44(X1,X2,X3)) -> mark#(X1)
p46: mark#(U44(X1,X2,X3)) -> active#(U44(mark(X1),X2,X3))
p47: active#(U94(tt(),V1,V2)) -> mark#(U95(isNat(V1),V2))
p48: mark#(U43(X1,X2,X3)) -> mark#(X1)
p49: mark#(U43(X1,X2,X3)) -> active#(U43(mark(X1),X2,X3))
p50: active#(U93(tt(),V1,V2)) -> mark#(U94(isNatIListKind(V2),V1,V2))
p51: mark#(U42(X1,X2,X3)) -> mark#(X1)
p52: mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3))
p53: active#(U92(tt(),V1,V2)) -> mark#(U93(isNatIListKind(V2),V1,V2))
p54: mark#(U41(X1,X2,X3)) -> mark#(X1)
p55: mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3))
p56: active#(U91(tt(),V1,V2)) -> mark#(U92(isNatKind(V1),V1,V2))
p57: mark#(U33(X)) -> mark#(X)
p58: mark#(U33(X)) -> active#(U33(mark(X)))
p59: active#(U61(tt(),V2)) -> mark#(U62(isNatIListKind(V2)))
p60: mark#(U32(X1,X2)) -> mark#(X1)
p61: mark#(U32(X1,X2)) -> active#(U32(mark(X1),X2))
p62: active#(U51(tt(),V2)) -> mark#(U52(isNatIListKind(V2)))
p63: mark#(U31(X1,X2)) -> mark#(X1)
p64: mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2))
p65: active#(U45(tt(),V2)) -> mark#(U46(isNatIList(V2)))
p66: mark#(U23(X)) -> mark#(X)
p67: mark#(U23(X)) -> active#(U23(mark(X)))
p68: active#(U44(tt(),V1,V2)) -> mark#(U45(isNat(V1),V2))
p69: mark#(U22(X1,X2)) -> mark#(X1)
p70: mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2))
p71: active#(U43(tt(),V1,V2)) -> mark#(U44(isNatIListKind(V2),V1,V2))
p72: mark#(U21(X1,X2)) -> mark#(X1)
p73: mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2))
p74: active#(U42(tt(),V1,V2)) -> mark#(U43(isNatIListKind(V2),V1,V2))
p75: mark#(take(X1,X2)) -> mark#(X2)
p76: mark#(take(X1,X2)) -> mark#(X1)
p77: mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2)))
p78: active#(U41(tt(),V1,V2)) -> mark#(U42(isNatKind(V1),V1,V2))
p79: mark#(U136(X1,X2,X3,X4)) -> mark#(X1)
p80: mark#(U136(X1,X2,X3,X4)) -> active#(U136(mark(X1),X2,X3,X4))
p81: active#(U32(tt(),V)) -> mark#(U33(isNatList(V)))
p82: mark#(U135(X1,X2,X3,X4)) -> mark#(X1)
p83: mark#(U135(X1,X2,X3,X4)) -> active#(U135(mark(X1),X2,X3,X4))
p84: active#(U31(tt(),V)) -> mark#(U32(isNatIListKind(V),V))
p85: mark#(U134(X1,X2,X3,X4)) -> mark#(X1)
p86: mark#(U134(X1,X2,X3,X4)) -> active#(U134(mark(X1),X2,X3,X4))
p87: active#(U22(tt(),V1)) -> mark#(U23(isNat(V1)))
p88: mark#(U133(X1,X2,X3,X4)) -> mark#(X1)
p89: mark#(U133(X1,X2,X3,X4)) -> active#(U133(mark(X1),X2,X3,X4))
p90: active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1))
p91: mark#(U132(X1,X2,X3,X4)) -> mark#(X1)
p92: mark#(U132(X1,X2,X3,X4)) -> active#(U132(mark(X1),X2,X3,X4))
p93: active#(U136(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL)))
p94: mark#(U131(X1,X2,X3,X4)) -> mark#(X1)
p95: mark#(U131(X1,X2,X3,X4)) -> active#(U131(mark(X1),X2,X3,X4))
p96: active#(U135(tt(),IL,M,N)) -> mark#(U136(isNatKind(N),IL,M,N))
p97: mark#(U122(X)) -> mark#(X)
p98: mark#(U122(X)) -> active#(U122(mark(X)))
p99: active#(U134(tt(),IL,M,N)) -> mark#(U135(isNat(N),IL,M,N))
p100: mark#(U121(X1,X2)) -> mark#(X1)
p101: mark#(U121(X1,X2)) -> active#(U121(mark(X1),X2))
p102: active#(U133(tt(),IL,M,N)) -> mark#(U134(isNatKind(M),IL,M,N))
p103: mark#(isNatList(X)) -> active#(isNatList(X))
p104: active#(U132(tt(),IL,M,N)) -> mark#(U133(isNat(M),IL,M,N))
p105: mark#(U13(X)) -> mark#(X)
p106: mark#(U13(X)) -> active#(U13(mark(X)))
p107: active#(U131(tt(),IL,M,N)) -> mark#(U132(isNatIListKind(IL),IL,M,N))
p108: mark#(length(X)) -> mark#(X)
p109: mark#(length(X)) -> active#(length(mark(X)))
p110: active#(U121(tt(),IL)) -> mark#(U122(isNatIListKind(IL)))
p111: mark#(s(X)) -> mark#(X)
p112: mark#(s(X)) -> active#(s(mark(X)))
p113: active#(U12(tt(),V1)) -> mark#(U13(isNatList(V1)))
p114: mark#(U114(X1,X2)) -> mark#(X1)
p115: mark#(U114(X1,X2)) -> active#(U114(mark(X1),X2))
p116: active#(U114(tt(),L)) -> mark#(s(length(L)))
p117: mark#(U113(X1,X2,X3)) -> mark#(X1)
p118: mark#(U113(X1,X2,X3)) -> active#(U113(mark(X1),X2,X3))
p119: active#(U113(tt(),L,N)) -> mark#(U114(isNatKind(N),L))
p120: mark#(U112(X1,X2,X3)) -> mark#(X1)
p121: mark#(U112(X1,X2,X3)) -> active#(U112(mark(X1),X2,X3))
p122: active#(U112(tt(),L,N)) -> mark#(U113(isNat(N),L,N))
p123: mark#(U111(X1,X2,X3)) -> mark#(X1)
p124: mark#(U111(X1,X2,X3)) -> active#(U111(mark(X1),X2,X3))
p125: active#(U111(tt(),L,N)) -> mark#(U112(isNatIListKind(L),L,N))
p126: mark#(U12(X1,X2)) -> mark#(X1)
p127: mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2))
p128: active#(U11(tt(),V1)) -> mark#(U12(isNatIListKind(V1),V1))
p129: mark#(U11(X1,X2)) -> mark#(X1)
p130: mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2))
p131: active#(U105(tt(),V2)) -> mark#(U106(isNatIList(V2)))
p132: mark#(isNatIList(X)) -> active#(isNatIList(X))
p133: active#(U104(tt(),V1,V2)) -> mark#(U105(isNat(V1),V2))
p134: mark#(U106(X)) -> mark#(X)
p135: mark#(U106(X)) -> active#(U106(mark(X)))
p136: active#(U103(tt(),V1,V2)) -> mark#(U104(isNatIListKind(V2),V1,V2))
p137: mark#(isNat(X)) -> active#(isNat(X))
p138: active#(U102(tt(),V1,V2)) -> mark#(U103(isNatIListKind(V2),V1,V2))
p139: mark#(U105(X1,X2)) -> mark#(X1)
p140: mark#(U105(X1,X2)) -> active#(U105(mark(X1),X2))
p141: active#(U101(tt(),V1,V2)) -> mark#(U102(isNatKind(V1),V1,V2))
p142: mark#(U104(X1,X2,X3)) -> mark#(X1)
p143: mark#(U104(X1,X2,X3)) -> active#(U104(mark(X1),X2,X3))
p144: mark#(isNatIListKind(X)) -> active#(isNatIListKind(X))
p145: mark#(U103(X1,X2,X3)) -> mark#(X1)
p146: mark#(U103(X1,X2,X3)) -> active#(U103(mark(X1),X2,X3))
p147: mark#(isNatKind(X)) -> active#(isNatKind(X))
p148: mark#(U102(X1,X2,X3)) -> mark#(X1)
p149: mark#(U102(X1,X2,X3)) -> active#(U102(mark(X1),X2,X3))
p150: mark#(U101(X1,X2,X3)) -> mark#(X1)
p151: mark#(U101(X1,X2,X3)) -> active#(U101(mark(X1),X2,X3))
p152: mark#(cons(X1,X2)) -> active#(cons(mark(X1),X2))
p153: mark#(zeros()) -> active#(zeros())

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41, r42, r43, r44, r45, r46, r47, r48, r49, r50, r51, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76, r77, r78, r79, r80, r81, r82, r83, r84, r85, r86, r87, r88, r89, r90, r91, r92, r93, r94, r95, r96, r97, r98, r99, r100, r101, r102, r103, r104, r105, r106, r107, r108, r109, r110, r111, r112, r113, r114, r115, r116, r117, r118, r119, r120, r121, r122, r123, r124, r125, r126, r127, r128, r129, r130, r131, r132, r133, r134, r135, r136, r137, r138, r139, r140, r141, r142, r143, r144, r145, r146, r147, r148, r149, r150, r151, r152, r153, r154, r155, r156, r157, r158, r159, r160, r161, r162, r163, r164, r165, r166, r167, r168, r169, r170, r171, r172, r173, r174, r175, r176, r177, r178, r179, r180, r181, r182, r183, r184, r185, r186, r187, r188, r189, r190, r191, r192, r193, r194, r195, r196, r197, r198, r199, r200, r201, r202, r203, r204, r205, r206, r207, r208, r209, r210, r211, r212, r213, r214, r215, r216, r217, r218, r219, r220, r221, r222, r223, r224, r225, r226, r227, r228, r229, r230, r231, r232, r233, r234, r235, r236, r237, r238, r239, r240, r241, r242, r243, r244, r245, r246, r247, r248, r249, r250, r251, r252, r253, r254, r255, r256, r257, r258, r259, r260, r261, r262, r263, r264, r265, r266, r267, r268, r269, r270, r271, r272, r273, r274, r275, r276, r277, r278, r279, r280, r281, r282, r283, r284, r285, r286, r287, r288, r289, r290, r291, r292, r293, r294, r295, r296, r297, r298, r299, r300, r301, r302, r303, r304, r305, r306, r307, r308, r309, r310, r311, r312, r313, r314, r315, r316, r317, r318, r319, r320, r321, r322, r323, r324, r325, r326, r327, r328, r329, r330, r331, r332, r333, r334, r335, r336, r337, r338, r339, r340, r341, r342, r343, r344, r345, r346, r347, r348, r349, r350, r351, r352, r353, r354, r355, r356, r357, r358

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    active#_A(x1) = max{0, x1 - 4}
    zeros_A = 89
    mark#_A(x1) = 85
    cons_A(x1,x2) = 76
    |0|_A = 61
    U96_A(x1) = 89
    mark_A(x1) = max{34, x1 - 10}
    take_A(x1,x2) = 89
    s_A(x1) = 89
    U131_A(x1,x2,x3,x4) = 89
    isNatIList_A(x1) = 89
    U95_A(x1,x2) = 89
    U121_A(x1,x2) = 89
    U94_A(x1,x2,x3) = 89
    length_A(x1) = 89
    U111_A(x1,x2,x3) = 89
    isNatList_A(x1) = 89
    U93_A(x1,x2,x3) = 89
    U101_A(x1,x2,x3) = 89
    isNatKind_A(x1) = 89
    U92_A(x1,x2,x3) = 89
    U91_A(x1,x2,x3) = 89
    U81_A(x1) = 89
    U71_A(x1) = 89
    isNatIListKind_A(x1) = 89
    U61_A(x1,x2) = 89
    U62_A(x1) = 89
    U51_A(x1,x2) = 89
    U41_A(x1,x2,x3) = 89
    U52_A(x1) = 89
    U31_A(x1,x2) = 89
    isNat_A(x1) = 89
    U21_A(x1,x2) = 89
    U46_A(x1) = 89
    U11_A(x1,x2) = 89
    U45_A(x1,x2) = 89
    tt_A = 46
    U44_A(x1,x2,x3) = 89
    U43_A(x1,x2,x3) = 89
    U42_A(x1,x2,x3) = 89
    U33_A(x1) = 89
    U32_A(x1,x2) = 89
    U23_A(x1) = 89
    U22_A(x1,x2) = 89
    U136_A(x1,x2,x3,x4) = 89
    U135_A(x1,x2,x3,x4) = 89
    U134_A(x1,x2,x3,x4) = 89
    U133_A(x1,x2,x3,x4) = 89
    U132_A(x1,x2,x3,x4) = 89
    U122_A(x1) = 89
    U13_A(x1) = 89
    U12_A(x1,x2) = 89
    U114_A(x1,x2) = 89
    U113_A(x1,x2,x3) = 89
    U112_A(x1,x2,x3) = 89
    U105_A(x1,x2) = 89
    U106_A(x1) = 89
    U104_A(x1,x2,x3) = 89
    U103_A(x1,x2,x3) = 89
    U102_A(x1,x2,x3) = 89
    active_A(x1) = max{35, x1 - 10}
    nil_A = 46

The next rules are strictly ordered:

  p152

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: active#(zeros()) -> mark#(cons(|0|(),zeros()))
p2: mark#(cons(X1,X2)) -> mark#(X1)
p3: mark#(U96(X)) -> mark#(X)
p4: mark#(U96(X)) -> active#(U96(mark(X)))
p5: active#(take(s(M),cons(N,IL))) -> mark#(U131(isNatIList(IL),IL,M,N))
p6: mark#(U95(X1,X2)) -> mark#(X1)
p7: mark#(U95(X1,X2)) -> active#(U95(mark(X1),X2))
p8: active#(take(|0|(),IL)) -> mark#(U121(isNatIList(IL),IL))
p9: mark#(U94(X1,X2,X3)) -> mark#(X1)
p10: mark#(U94(X1,X2,X3)) -> active#(U94(mark(X1),X2,X3))
p11: active#(length(cons(N,L))) -> mark#(U111(isNatList(L),L,N))
p12: mark#(U93(X1,X2,X3)) -> mark#(X1)
p13: mark#(U93(X1,X2,X3)) -> active#(U93(mark(X1),X2,X3))
p14: active#(isNatList(take(V1,V2))) -> mark#(U101(isNatKind(V1),V1,V2))
p15: mark#(U92(X1,X2,X3)) -> mark#(X1)
p16: mark#(U92(X1,X2,X3)) -> active#(U92(mark(X1),X2,X3))
p17: active#(isNatList(cons(V1,V2))) -> mark#(U91(isNatKind(V1),V1,V2))
p18: mark#(U91(X1,X2,X3)) -> mark#(X1)
p19: mark#(U91(X1,X2,X3)) -> active#(U91(mark(X1),X2,X3))
p20: active#(isNatKind(s(V1))) -> mark#(U81(isNatKind(V1)))
p21: mark#(U81(X)) -> mark#(X)
p22: mark#(U81(X)) -> active#(U81(mark(X)))
p23: active#(isNatKind(length(V1))) -> mark#(U71(isNatIListKind(V1)))
p24: mark#(U71(X)) -> mark#(X)
p25: mark#(U71(X)) -> active#(U71(mark(X)))
p26: active#(isNatIListKind(take(V1,V2))) -> mark#(U61(isNatKind(V1),V2))
p27: mark#(U62(X)) -> mark#(X)
p28: mark#(U62(X)) -> active#(U62(mark(X)))
p29: active#(isNatIListKind(cons(V1,V2))) -> mark#(U51(isNatKind(V1),V2))
p30: mark#(U61(X1,X2)) -> mark#(X1)
p31: mark#(U61(X1,X2)) -> active#(U61(mark(X1),X2))
p32: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNatKind(V1),V1,V2))
p33: mark#(U52(X)) -> mark#(X)
p34: mark#(U52(X)) -> active#(U52(mark(X)))
p35: active#(isNatIList(V)) -> mark#(U31(isNatIListKind(V),V))
p36: mark#(U51(X1,X2)) -> mark#(X1)
p37: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p38: active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1))
p39: mark#(U46(X)) -> mark#(X)
p40: mark#(U46(X)) -> active#(U46(mark(X)))
p41: active#(isNat(length(V1))) -> mark#(U11(isNatIListKind(V1),V1))
p42: mark#(U45(X1,X2)) -> mark#(X1)
p43: mark#(U45(X1,X2)) -> active#(U45(mark(X1),X2))
p44: active#(U95(tt(),V2)) -> mark#(U96(isNatList(V2)))
p45: mark#(U44(X1,X2,X3)) -> mark#(X1)
p46: mark#(U44(X1,X2,X3)) -> active#(U44(mark(X1),X2,X3))
p47: active#(U94(tt(),V1,V2)) -> mark#(U95(isNat(V1),V2))
p48: mark#(U43(X1,X2,X3)) -> mark#(X1)
p49: mark#(U43(X1,X2,X3)) -> active#(U43(mark(X1),X2,X3))
p50: active#(U93(tt(),V1,V2)) -> mark#(U94(isNatIListKind(V2),V1,V2))
p51: mark#(U42(X1,X2,X3)) -> mark#(X1)
p52: mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3))
p53: active#(U92(tt(),V1,V2)) -> mark#(U93(isNatIListKind(V2),V1,V2))
p54: mark#(U41(X1,X2,X3)) -> mark#(X1)
p55: mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3))
p56: active#(U91(tt(),V1,V2)) -> mark#(U92(isNatKind(V1),V1,V2))
p57: mark#(U33(X)) -> mark#(X)
p58: mark#(U33(X)) -> active#(U33(mark(X)))
p59: active#(U61(tt(),V2)) -> mark#(U62(isNatIListKind(V2)))
p60: mark#(U32(X1,X2)) -> mark#(X1)
p61: mark#(U32(X1,X2)) -> active#(U32(mark(X1),X2))
p62: active#(U51(tt(),V2)) -> mark#(U52(isNatIListKind(V2)))
p63: mark#(U31(X1,X2)) -> mark#(X1)
p64: mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2))
p65: active#(U45(tt(),V2)) -> mark#(U46(isNatIList(V2)))
p66: mark#(U23(X)) -> mark#(X)
p67: mark#(U23(X)) -> active#(U23(mark(X)))
p68: active#(U44(tt(),V1,V2)) -> mark#(U45(isNat(V1),V2))
p69: mark#(U22(X1,X2)) -> mark#(X1)
p70: mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2))
p71: active#(U43(tt(),V1,V2)) -> mark#(U44(isNatIListKind(V2),V1,V2))
p72: mark#(U21(X1,X2)) -> mark#(X1)
p73: mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2))
p74: active#(U42(tt(),V1,V2)) -> mark#(U43(isNatIListKind(V2),V1,V2))
p75: mark#(take(X1,X2)) -> mark#(X2)
p76: mark#(take(X1,X2)) -> mark#(X1)
p77: mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2)))
p78: active#(U41(tt(),V1,V2)) -> mark#(U42(isNatKind(V1),V1,V2))
p79: mark#(U136(X1,X2,X3,X4)) -> mark#(X1)
p80: mark#(U136(X1,X2,X3,X4)) -> active#(U136(mark(X1),X2,X3,X4))
p81: active#(U32(tt(),V)) -> mark#(U33(isNatList(V)))
p82: mark#(U135(X1,X2,X3,X4)) -> mark#(X1)
p83: mark#(U135(X1,X2,X3,X4)) -> active#(U135(mark(X1),X2,X3,X4))
p84: active#(U31(tt(),V)) -> mark#(U32(isNatIListKind(V),V))
p85: mark#(U134(X1,X2,X3,X4)) -> mark#(X1)
p86: mark#(U134(X1,X2,X3,X4)) -> active#(U134(mark(X1),X2,X3,X4))
p87: active#(U22(tt(),V1)) -> mark#(U23(isNat(V1)))
p88: mark#(U133(X1,X2,X3,X4)) -> mark#(X1)
p89: mark#(U133(X1,X2,X3,X4)) -> active#(U133(mark(X1),X2,X3,X4))
p90: active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1))
p91: mark#(U132(X1,X2,X3,X4)) -> mark#(X1)
p92: mark#(U132(X1,X2,X3,X4)) -> active#(U132(mark(X1),X2,X3,X4))
p93: active#(U136(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL)))
p94: mark#(U131(X1,X2,X3,X4)) -> mark#(X1)
p95: mark#(U131(X1,X2,X3,X4)) -> active#(U131(mark(X1),X2,X3,X4))
p96: active#(U135(tt(),IL,M,N)) -> mark#(U136(isNatKind(N),IL,M,N))
p97: mark#(U122(X)) -> mark#(X)
p98: mark#(U122(X)) -> active#(U122(mark(X)))
p99: active#(U134(tt(),IL,M,N)) -> mark#(U135(isNat(N),IL,M,N))
p100: mark#(U121(X1,X2)) -> mark#(X1)
p101: mark#(U121(X1,X2)) -> active#(U121(mark(X1),X2))
p102: active#(U133(tt(),IL,M,N)) -> mark#(U134(isNatKind(M),IL,M,N))
p103: mark#(isNatList(X)) -> active#(isNatList(X))
p104: active#(U132(tt(),IL,M,N)) -> mark#(U133(isNat(M),IL,M,N))
p105: mark#(U13(X)) -> mark#(X)
p106: mark#(U13(X)) -> active#(U13(mark(X)))
p107: active#(U131(tt(),IL,M,N)) -> mark#(U132(isNatIListKind(IL),IL,M,N))
p108: mark#(length(X)) -> mark#(X)
p109: mark#(length(X)) -> active#(length(mark(X)))
p110: active#(U121(tt(),IL)) -> mark#(U122(isNatIListKind(IL)))
p111: mark#(s(X)) -> mark#(X)
p112: mark#(s(X)) -> active#(s(mark(X)))
p113: active#(U12(tt(),V1)) -> mark#(U13(isNatList(V1)))
p114: mark#(U114(X1,X2)) -> mark#(X1)
p115: mark#(U114(X1,X2)) -> active#(U114(mark(X1),X2))
p116: active#(U114(tt(),L)) -> mark#(s(length(L)))
p117: mark#(U113(X1,X2,X3)) -> mark#(X1)
p118: mark#(U113(X1,X2,X3)) -> active#(U113(mark(X1),X2,X3))
p119: active#(U113(tt(),L,N)) -> mark#(U114(isNatKind(N),L))
p120: mark#(U112(X1,X2,X3)) -> mark#(X1)
p121: mark#(U112(X1,X2,X3)) -> active#(U112(mark(X1),X2,X3))
p122: active#(U112(tt(),L,N)) -> mark#(U113(isNat(N),L,N))
p123: mark#(U111(X1,X2,X3)) -> mark#(X1)
p124: mark#(U111(X1,X2,X3)) -> active#(U111(mark(X1),X2,X3))
p125: active#(U111(tt(),L,N)) -> mark#(U112(isNatIListKind(L),L,N))
p126: mark#(U12(X1,X2)) -> mark#(X1)
p127: mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2))
p128: active#(U11(tt(),V1)) -> mark#(U12(isNatIListKind(V1),V1))
p129: mark#(U11(X1,X2)) -> mark#(X1)
p130: mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2))
p131: active#(U105(tt(),V2)) -> mark#(U106(isNatIList(V2)))
p132: mark#(isNatIList(X)) -> active#(isNatIList(X))
p133: active#(U104(tt(),V1,V2)) -> mark#(U105(isNat(V1),V2))
p134: mark#(U106(X)) -> mark#(X)
p135: mark#(U106(X)) -> active#(U106(mark(X)))
p136: active#(U103(tt(),V1,V2)) -> mark#(U104(isNatIListKind(V2),V1,V2))
p137: mark#(isNat(X)) -> active#(isNat(X))
p138: active#(U102(tt(),V1,V2)) -> mark#(U103(isNatIListKind(V2),V1,V2))
p139: mark#(U105(X1,X2)) -> mark#(X1)
p140: mark#(U105(X1,X2)) -> active#(U105(mark(X1),X2))
p141: active#(U101(tt(),V1,V2)) -> mark#(U102(isNatKind(V1),V1,V2))
p142: mark#(U104(X1,X2,X3)) -> mark#(X1)
p143: mark#(U104(X1,X2,X3)) -> active#(U104(mark(X1),X2,X3))
p144: mark#(isNatIListKind(X)) -> active#(isNatIListKind(X))
p145: mark#(U103(X1,X2,X3)) -> mark#(X1)
p146: mark#(U103(X1,X2,X3)) -> active#(U103(mark(X1),X2,X3))
p147: mark#(isNatKind(X)) -> active#(isNatKind(X))
p148: mark#(U102(X1,X2,X3)) -> mark#(X1)
p149: mark#(U102(X1,X2,X3)) -> active#(U102(mark(X1),X2,X3))
p150: mark#(U101(X1,X2,X3)) -> mark#(X1)
p151: mark#(U101(X1,X2,X3)) -> active#(U101(mark(X1),X2,X3))
p152: mark#(zeros()) -> active#(zeros())

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19, p20, p21, p22, p23, p24, p25, p26, p27, p28, p29, p30, p31, p32, p33, p34, p35, p36, p37, p38, p39, p40, p41, p42, p43, p44, p45, p46, p47, p48, p49, p50, p51, p52, p53, p54, p55, p56, p57, p58, p59, p60, p61, p62, p63, p64, p65, p66, p67, p68, p69, p70, p71, p72, p73, p74, p75, p76, p77, p78, p79, p80, p81, p82, p83, p84, p85, p86, p87, p88, p89, p90, p91, p92, p93, p94, p95, p96, p97, p98, p99, p100, p101, p102, p103, p104, p105, p106, p107, p108, p109, p110, p111, p112, p113, p114, p115, p116, p117, p118, p119, p120, p121, p122, p123, p124, p125, p126, p127, p128, p129, p130, p131, p132, p133, p134, p135, p136, p137, p138, p139, p140, p141, p142, p143, p144, p145, p146, p147, p148, p149, p150, p151, p152}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: active#(zeros()) -> mark#(cons(|0|(),zeros()))
p2: mark#(cons(X1,X2)) -> mark#(X1)
p3: mark#(zeros()) -> active#(zeros())
p4: mark#(U101(X1,X2,X3)) -> active#(U101(mark(X1),X2,X3))
p5: active#(U101(tt(),V1,V2)) -> mark#(U102(isNatKind(V1),V1,V2))
p6: mark#(U101(X1,X2,X3)) -> mark#(X1)
p7: mark#(U102(X1,X2,X3)) -> active#(U102(mark(X1),X2,X3))
p8: active#(U102(tt(),V1,V2)) -> mark#(U103(isNatIListKind(V2),V1,V2))
p9: mark#(U102(X1,X2,X3)) -> mark#(X1)
p10: mark#(isNatKind(X)) -> active#(isNatKind(X))
p11: active#(U103(tt(),V1,V2)) -> mark#(U104(isNatIListKind(V2),V1,V2))
p12: mark#(U103(X1,X2,X3)) -> active#(U103(mark(X1),X2,X3))
p13: active#(U104(tt(),V1,V2)) -> mark#(U105(isNat(V1),V2))
p14: mark#(U103(X1,X2,X3)) -> mark#(X1)
p15: mark#(isNatIListKind(X)) -> active#(isNatIListKind(X))
p16: active#(U105(tt(),V2)) -> mark#(U106(isNatIList(V2)))
p17: mark#(U104(X1,X2,X3)) -> active#(U104(mark(X1),X2,X3))
p18: active#(U11(tt(),V1)) -> mark#(U12(isNatIListKind(V1),V1))
p19: mark#(U104(X1,X2,X3)) -> mark#(X1)
p20: mark#(U105(X1,X2)) -> active#(U105(mark(X1),X2))
p21: active#(U111(tt(),L,N)) -> mark#(U112(isNatIListKind(L),L,N))
p22: mark#(U105(X1,X2)) -> mark#(X1)
p23: mark#(isNat(X)) -> active#(isNat(X))
p24: active#(U112(tt(),L,N)) -> mark#(U113(isNat(N),L,N))
p25: mark#(U106(X)) -> active#(U106(mark(X)))
p26: active#(U113(tt(),L,N)) -> mark#(U114(isNatKind(N),L))
p27: mark#(U106(X)) -> mark#(X)
p28: mark#(isNatIList(X)) -> active#(isNatIList(X))
p29: active#(U114(tt(),L)) -> mark#(s(length(L)))
p30: mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2))
p31: active#(U12(tt(),V1)) -> mark#(U13(isNatList(V1)))
p32: mark#(U11(X1,X2)) -> mark#(X1)
p33: mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2))
p34: active#(U121(tt(),IL)) -> mark#(U122(isNatIListKind(IL)))
p35: mark#(U12(X1,X2)) -> mark#(X1)
p36: mark#(U111(X1,X2,X3)) -> active#(U111(mark(X1),X2,X3))
p37: active#(U131(tt(),IL,M,N)) -> mark#(U132(isNatIListKind(IL),IL,M,N))
p38: mark#(U111(X1,X2,X3)) -> mark#(X1)
p39: mark#(U112(X1,X2,X3)) -> active#(U112(mark(X1),X2,X3))
p40: active#(U132(tt(),IL,M,N)) -> mark#(U133(isNat(M),IL,M,N))
p41: mark#(U112(X1,X2,X3)) -> mark#(X1)
p42: mark#(U113(X1,X2,X3)) -> active#(U113(mark(X1),X2,X3))
p43: active#(U133(tt(),IL,M,N)) -> mark#(U134(isNatKind(M),IL,M,N))
p44: mark#(U113(X1,X2,X3)) -> mark#(X1)
p45: mark#(U114(X1,X2)) -> active#(U114(mark(X1),X2))
p46: active#(U134(tt(),IL,M,N)) -> mark#(U135(isNat(N),IL,M,N))
p47: mark#(U114(X1,X2)) -> mark#(X1)
p48: mark#(s(X)) -> active#(s(mark(X)))
p49: active#(U135(tt(),IL,M,N)) -> mark#(U136(isNatKind(N),IL,M,N))
p50: mark#(s(X)) -> mark#(X)
p51: mark#(length(X)) -> active#(length(mark(X)))
p52: active#(U136(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL)))
p53: mark#(length(X)) -> mark#(X)
p54: mark#(U13(X)) -> active#(U13(mark(X)))
p55: active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1))
p56: mark#(U13(X)) -> mark#(X)
p57: mark#(isNatList(X)) -> active#(isNatList(X))
p58: active#(U22(tt(),V1)) -> mark#(U23(isNat(V1)))
p59: mark#(U121(X1,X2)) -> active#(U121(mark(X1),X2))
p60: active#(U31(tt(),V)) -> mark#(U32(isNatIListKind(V),V))
p61: mark#(U121(X1,X2)) -> mark#(X1)
p62: mark#(U122(X)) -> active#(U122(mark(X)))
p63: active#(U32(tt(),V)) -> mark#(U33(isNatList(V)))
p64: mark#(U122(X)) -> mark#(X)
p65: mark#(U131(X1,X2,X3,X4)) -> active#(U131(mark(X1),X2,X3,X4))
p66: active#(U41(tt(),V1,V2)) -> mark#(U42(isNatKind(V1),V1,V2))
p67: mark#(U131(X1,X2,X3,X4)) -> mark#(X1)
p68: mark#(U132(X1,X2,X3,X4)) -> active#(U132(mark(X1),X2,X3,X4))
p69: active#(U42(tt(),V1,V2)) -> mark#(U43(isNatIListKind(V2),V1,V2))
p70: mark#(U132(X1,X2,X3,X4)) -> mark#(X1)
p71: mark#(U133(X1,X2,X3,X4)) -> active#(U133(mark(X1),X2,X3,X4))
p72: active#(U43(tt(),V1,V2)) -> mark#(U44(isNatIListKind(V2),V1,V2))
p73: mark#(U133(X1,X2,X3,X4)) -> mark#(X1)
p74: mark#(U134(X1,X2,X3,X4)) -> active#(U134(mark(X1),X2,X3,X4))
p75: active#(U44(tt(),V1,V2)) -> mark#(U45(isNat(V1),V2))
p76: mark#(U134(X1,X2,X3,X4)) -> mark#(X1)
p77: mark#(U135(X1,X2,X3,X4)) -> active#(U135(mark(X1),X2,X3,X4))
p78: active#(U45(tt(),V2)) -> mark#(U46(isNatIList(V2)))
p79: mark#(U135(X1,X2,X3,X4)) -> mark#(X1)
p80: mark#(U136(X1,X2,X3,X4)) -> active#(U136(mark(X1),X2,X3,X4))
p81: active#(U51(tt(),V2)) -> mark#(U52(isNatIListKind(V2)))
p82: mark#(U136(X1,X2,X3,X4)) -> mark#(X1)
p83: mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2)))
p84: active#(U61(tt(),V2)) -> mark#(U62(isNatIListKind(V2)))
p85: mark#(take(X1,X2)) -> mark#(X1)
p86: mark#(take(X1,X2)) -> mark#(X2)
p87: mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2))
p88: active#(U91(tt(),V1,V2)) -> mark#(U92(isNatKind(V1),V1,V2))
p89: mark#(U21(X1,X2)) -> mark#(X1)
p90: mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2))
p91: active#(U92(tt(),V1,V2)) -> mark#(U93(isNatIListKind(V2),V1,V2))
p92: mark#(U22(X1,X2)) -> mark#(X1)
p93: mark#(U23(X)) -> active#(U23(mark(X)))
p94: active#(U93(tt(),V1,V2)) -> mark#(U94(isNatIListKind(V2),V1,V2))
p95: mark#(U23(X)) -> mark#(X)
p96: mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2))
p97: active#(U94(tt(),V1,V2)) -> mark#(U95(isNat(V1),V2))
p98: mark#(U31(X1,X2)) -> mark#(X1)
p99: mark#(U32(X1,X2)) -> active#(U32(mark(X1),X2))
p100: active#(U95(tt(),V2)) -> mark#(U96(isNatList(V2)))
p101: mark#(U32(X1,X2)) -> mark#(X1)
p102: mark#(U33(X)) -> active#(U33(mark(X)))
p103: active#(isNat(length(V1))) -> mark#(U11(isNatIListKind(V1),V1))
p104: mark#(U33(X)) -> mark#(X)
p105: mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3))
p106: active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1))
p107: mark#(U41(X1,X2,X3)) -> mark#(X1)
p108: mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3))
p109: active#(isNatIList(V)) -> mark#(U31(isNatIListKind(V),V))
p110: mark#(U42(X1,X2,X3)) -> mark#(X1)
p111: mark#(U43(X1,X2,X3)) -> active#(U43(mark(X1),X2,X3))
p112: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNatKind(V1),V1,V2))
p113: mark#(U43(X1,X2,X3)) -> mark#(X1)
p114: mark#(U44(X1,X2,X3)) -> active#(U44(mark(X1),X2,X3))
p115: active#(isNatIListKind(cons(V1,V2))) -> mark#(U51(isNatKind(V1),V2))
p116: mark#(U44(X1,X2,X3)) -> mark#(X1)
p117: mark#(U45(X1,X2)) -> active#(U45(mark(X1),X2))
p118: active#(isNatIListKind(take(V1,V2))) -> mark#(U61(isNatKind(V1),V2))
p119: mark#(U45(X1,X2)) -> mark#(X1)
p120: mark#(U46(X)) -> active#(U46(mark(X)))
p121: active#(isNatKind(length(V1))) -> mark#(U71(isNatIListKind(V1)))
p122: mark#(U46(X)) -> mark#(X)
p123: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p124: active#(isNatKind(s(V1))) -> mark#(U81(isNatKind(V1)))
p125: mark#(U51(X1,X2)) -> mark#(X1)
p126: mark#(U52(X)) -> active#(U52(mark(X)))
p127: active#(isNatList(cons(V1,V2))) -> mark#(U91(isNatKind(V1),V1,V2))
p128: mark#(U52(X)) -> mark#(X)
p129: mark#(U61(X1,X2)) -> active#(U61(mark(X1),X2))
p130: active#(isNatList(take(V1,V2))) -> mark#(U101(isNatKind(V1),V1,V2))
p131: mark#(U61(X1,X2)) -> mark#(X1)
p132: mark#(U62(X)) -> active#(U62(mark(X)))
p133: active#(length(cons(N,L))) -> mark#(U111(isNatList(L),L,N))
p134: mark#(U62(X)) -> mark#(X)
p135: mark#(U71(X)) -> active#(U71(mark(X)))
p136: active#(take(|0|(),IL)) -> mark#(U121(isNatIList(IL),IL))
p137: mark#(U71(X)) -> mark#(X)
p138: mark#(U81(X)) -> active#(U81(mark(X)))
p139: active#(take(s(M),cons(N,IL))) -> mark#(U131(isNatIList(IL),IL,M,N))
p140: mark#(U81(X)) -> mark#(X)
p141: mark#(U91(X1,X2,X3)) -> active#(U91(mark(X1),X2,X3))
p142: mark#(U91(X1,X2,X3)) -> mark#(X1)
p143: mark#(U92(X1,X2,X3)) -> active#(U92(mark(X1),X2,X3))
p144: mark#(U92(X1,X2,X3)) -> mark#(X1)
p145: mark#(U93(X1,X2,X3)) -> active#(U93(mark(X1),X2,X3))
p146: mark#(U93(X1,X2,X3)) -> mark#(X1)
p147: mark#(U94(X1,X2,X3)) -> active#(U94(mark(X1),X2,X3))
p148: mark#(U94(X1,X2,X3)) -> mark#(X1)
p149: mark#(U95(X1,X2)) -> active#(U95(mark(X1),X2))
p150: mark#(U95(X1,X2)) -> mark#(X1)
p151: mark#(U96(X)) -> active#(U96(mark(X)))
p152: mark#(U96(X)) -> mark#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41, r42, r43, r44, r45, r46, r47, r48, r49, r50, r51, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76, r77, r78, r79, r80, r81, r82, r83, r84, r85, r86, r87, r88, r89, r90, r91, r92, r93, r94, r95, r96, r97, r98, r99, r100, r101, r102, r103, r104, r105, r106, r107, r108, r109, r110, r111, r112, r113, r114, r115, r116, r117, r118, r119, r120, r121, r122, r123, r124, r125, r126, r127, r128, r129, r130, r131, r132, r133, r134, r135, r136, r137, r138, r139, r140, r141, r142, r143, r144, r145, r146, r147, r148, r149, r150, r151, r152, r153, r154, r155, r156, r157, r158, r159, r160, r161, r162, r163, r164, r165, r166, r167, r168, r169, r170, r171, r172, r173, r174, r175, r176, r177, r178, r179, r180, r181, r182, r183, r184, r185, r186, r187, r188, r189, r190, r191, r192, r193, r194, r195, r196, r197, r198, r199, r200, r201, r202, r203, r204, r205, r206, r207, r208, r209, r210, r211, r212, r213, r214, r215, r216, r217, r218, r219, r220, r221, r222, r223, r224, r225, r226, r227, r228, r229, r230, r231, r232, r233, r234, r235, r236, r237, r238, r239, r240, r241, r242, r243, r244, r245, r246, r247, r248, r249, r250, r251, r252, r253, r254, r255, r256, r257, r258, r259, r260, r261, r262, r263, r264, r265, r266, r267, r268, r269, r270, r271, r272, r273, r274, r275, r276, r277, r278, r279, r280, r281, r282, r283, r284, r285, r286, r287, r288, r289, r290, r291, r292, r293, r294, r295, r296, r297, r298, r299, r300, r301, r302, r303, r304, r305, r306, r307, r308, r309, r310, r311, r312, r313, r314, r315, r316, r317, r318, r319, r320, r321, r322, r323, r324, r325, r326, r327, r328, r329, r330, r331, r332, r333, r334, r335, r336, r337, r338, r339, r340, r341, r342, r343, r344, r345, r346, r347, r348, r349, r350, r351, r352, r353, r354, r355, r356, r357, r358

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    active#_A(x1) = max{0, x1 - 364}
    zeros_A = 381
    mark#_A(x1) = 17
    cons_A(x1,x2) = 351
    |0|_A = 341
    U101_A(x1,x2,x3) = 381
    mark_A(x1) = x1 + 426
    tt_A = 10
    U102_A(x1,x2,x3) = 381
    isNatKind_A(x1) = 381
    U103_A(x1,x2,x3) = 381
    isNatIListKind_A(x1) = 381
    U104_A(x1,x2,x3) = 381
    U105_A(x1,x2) = 381
    isNat_A(x1) = 381
    U106_A(x1) = 380
    isNatIList_A(x1) = 381
    U11_A(x1,x2) = 381
    U12_A(x1,x2) = 381
    U111_A(x1,x2,x3) = 381
    U112_A(x1,x2,x3) = 381
    U113_A(x1,x2,x3) = 381
    U114_A(x1,x2) = 381
    s_A(x1) = 381
    length_A(x1) = 381
    U13_A(x1) = 333
    isNatList_A(x1) = 381
    U121_A(x1,x2) = 381
    U122_A(x1) = 352
    U131_A(x1,x2,x3,x4) = 381
    U132_A(x1,x2,x3,x4) = 381
    U133_A(x1,x2,x3,x4) = 381
    U134_A(x1,x2,x3,x4) = 381
    U135_A(x1,x2,x3,x4) = 381
    U136_A(x1,x2,x3,x4) = 381
    take_A(x1,x2) = 381
    U21_A(x1,x2) = 381
    U22_A(x1,x2) = 381
    U23_A(x1) = 351
    U31_A(x1,x2) = 381
    U32_A(x1,x2) = 381
    U33_A(x1) = 317
    U41_A(x1,x2,x3) = 381
    U42_A(x1,x2,x3) = 381
    U43_A(x1,x2,x3) = 381
    U44_A(x1,x2,x3) = 381
    U45_A(x1,x2) = 381
    U46_A(x1) = 381
    U51_A(x1,x2) = 381
    U52_A(x1) = 381
    U61_A(x1,x2) = 381
    U62_A(x1) = 381
    U91_A(x1,x2,x3) = 381
    U92_A(x1,x2,x3) = 381
    U93_A(x1,x2,x3) = 381
    U94_A(x1,x2,x3) = 381
    U95_A(x1,x2) = 381
    U96_A(x1) = 10
    U71_A(x1) = 30
    U81_A(x1) = 381
    active_A(x1) = max{435, x1 + 426}
    nil_A = 10

The next rules are strictly ordered:

  p25, p54, p62, p93, p102, p135, p151

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: active#(zeros()) -> mark#(cons(|0|(),zeros()))
p2: mark#(cons(X1,X2)) -> mark#(X1)
p3: mark#(zeros()) -> active#(zeros())
p4: mark#(U101(X1,X2,X3)) -> active#(U101(mark(X1),X2,X3))
p5: active#(U101(tt(),V1,V2)) -> mark#(U102(isNatKind(V1),V1,V2))
p6: mark#(U101(X1,X2,X3)) -> mark#(X1)
p7: mark#(U102(X1,X2,X3)) -> active#(U102(mark(X1),X2,X3))
p8: active#(U102(tt(),V1,V2)) -> mark#(U103(isNatIListKind(V2),V1,V2))
p9: mark#(U102(X1,X2,X3)) -> mark#(X1)
p10: mark#(isNatKind(X)) -> active#(isNatKind(X))
p11: active#(U103(tt(),V1,V2)) -> mark#(U104(isNatIListKind(V2),V1,V2))
p12: mark#(U103(X1,X2,X3)) -> active#(U103(mark(X1),X2,X3))
p13: active#(U104(tt(),V1,V2)) -> mark#(U105(isNat(V1),V2))
p14: mark#(U103(X1,X2,X3)) -> mark#(X1)
p15: mark#(isNatIListKind(X)) -> active#(isNatIListKind(X))
p16: active#(U105(tt(),V2)) -> mark#(U106(isNatIList(V2)))
p17: mark#(U104(X1,X2,X3)) -> active#(U104(mark(X1),X2,X3))
p18: active#(U11(tt(),V1)) -> mark#(U12(isNatIListKind(V1),V1))
p19: mark#(U104(X1,X2,X3)) -> mark#(X1)
p20: mark#(U105(X1,X2)) -> active#(U105(mark(X1),X2))
p21: active#(U111(tt(),L,N)) -> mark#(U112(isNatIListKind(L),L,N))
p22: mark#(U105(X1,X2)) -> mark#(X1)
p23: mark#(isNat(X)) -> active#(isNat(X))
p24: active#(U112(tt(),L,N)) -> mark#(U113(isNat(N),L,N))
p25: active#(U113(tt(),L,N)) -> mark#(U114(isNatKind(N),L))
p26: mark#(U106(X)) -> mark#(X)
p27: mark#(isNatIList(X)) -> active#(isNatIList(X))
p28: active#(U114(tt(),L)) -> mark#(s(length(L)))
p29: mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2))
p30: active#(U12(tt(),V1)) -> mark#(U13(isNatList(V1)))
p31: mark#(U11(X1,X2)) -> mark#(X1)
p32: mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2))
p33: active#(U121(tt(),IL)) -> mark#(U122(isNatIListKind(IL)))
p34: mark#(U12(X1,X2)) -> mark#(X1)
p35: mark#(U111(X1,X2,X3)) -> active#(U111(mark(X1),X2,X3))
p36: active#(U131(tt(),IL,M,N)) -> mark#(U132(isNatIListKind(IL),IL,M,N))
p37: mark#(U111(X1,X2,X3)) -> mark#(X1)
p38: mark#(U112(X1,X2,X3)) -> active#(U112(mark(X1),X2,X3))
p39: active#(U132(tt(),IL,M,N)) -> mark#(U133(isNat(M),IL,M,N))
p40: mark#(U112(X1,X2,X3)) -> mark#(X1)
p41: mark#(U113(X1,X2,X3)) -> active#(U113(mark(X1),X2,X3))
p42: active#(U133(tt(),IL,M,N)) -> mark#(U134(isNatKind(M),IL,M,N))
p43: mark#(U113(X1,X2,X3)) -> mark#(X1)
p44: mark#(U114(X1,X2)) -> active#(U114(mark(X1),X2))
p45: active#(U134(tt(),IL,M,N)) -> mark#(U135(isNat(N),IL,M,N))
p46: mark#(U114(X1,X2)) -> mark#(X1)
p47: mark#(s(X)) -> active#(s(mark(X)))
p48: active#(U135(tt(),IL,M,N)) -> mark#(U136(isNatKind(N),IL,M,N))
p49: mark#(s(X)) -> mark#(X)
p50: mark#(length(X)) -> active#(length(mark(X)))
p51: active#(U136(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL)))
p52: mark#(length(X)) -> mark#(X)
p53: active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1))
p54: mark#(U13(X)) -> mark#(X)
p55: mark#(isNatList(X)) -> active#(isNatList(X))
p56: active#(U22(tt(),V1)) -> mark#(U23(isNat(V1)))
p57: mark#(U121(X1,X2)) -> active#(U121(mark(X1),X2))
p58: active#(U31(tt(),V)) -> mark#(U32(isNatIListKind(V),V))
p59: mark#(U121(X1,X2)) -> mark#(X1)
p60: active#(U32(tt(),V)) -> mark#(U33(isNatList(V)))
p61: mark#(U122(X)) -> mark#(X)
p62: mark#(U131(X1,X2,X3,X4)) -> active#(U131(mark(X1),X2,X3,X4))
p63: active#(U41(tt(),V1,V2)) -> mark#(U42(isNatKind(V1),V1,V2))
p64: mark#(U131(X1,X2,X3,X4)) -> mark#(X1)
p65: mark#(U132(X1,X2,X3,X4)) -> active#(U132(mark(X1),X2,X3,X4))
p66: active#(U42(tt(),V1,V2)) -> mark#(U43(isNatIListKind(V2),V1,V2))
p67: mark#(U132(X1,X2,X3,X4)) -> mark#(X1)
p68: mark#(U133(X1,X2,X3,X4)) -> active#(U133(mark(X1),X2,X3,X4))
p69: active#(U43(tt(),V1,V2)) -> mark#(U44(isNatIListKind(V2),V1,V2))
p70: mark#(U133(X1,X2,X3,X4)) -> mark#(X1)
p71: mark#(U134(X1,X2,X3,X4)) -> active#(U134(mark(X1),X2,X3,X4))
p72: active#(U44(tt(),V1,V2)) -> mark#(U45(isNat(V1),V2))
p73: mark#(U134(X1,X2,X3,X4)) -> mark#(X1)
p74: mark#(U135(X1,X2,X3,X4)) -> active#(U135(mark(X1),X2,X3,X4))
p75: active#(U45(tt(),V2)) -> mark#(U46(isNatIList(V2)))
p76: mark#(U135(X1,X2,X3,X4)) -> mark#(X1)
p77: mark#(U136(X1,X2,X3,X4)) -> active#(U136(mark(X1),X2,X3,X4))
p78: active#(U51(tt(),V2)) -> mark#(U52(isNatIListKind(V2)))
p79: mark#(U136(X1,X2,X3,X4)) -> mark#(X1)
p80: mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2)))
p81: active#(U61(tt(),V2)) -> mark#(U62(isNatIListKind(V2)))
p82: mark#(take(X1,X2)) -> mark#(X1)
p83: mark#(take(X1,X2)) -> mark#(X2)
p84: mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2))
p85: active#(U91(tt(),V1,V2)) -> mark#(U92(isNatKind(V1),V1,V2))
p86: mark#(U21(X1,X2)) -> mark#(X1)
p87: mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2))
p88: active#(U92(tt(),V1,V2)) -> mark#(U93(isNatIListKind(V2),V1,V2))
p89: mark#(U22(X1,X2)) -> mark#(X1)
p90: active#(U93(tt(),V1,V2)) -> mark#(U94(isNatIListKind(V2),V1,V2))
p91: mark#(U23(X)) -> mark#(X)
p92: mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2))
p93: active#(U94(tt(),V1,V2)) -> mark#(U95(isNat(V1),V2))
p94: mark#(U31(X1,X2)) -> mark#(X1)
p95: mark#(U32(X1,X2)) -> active#(U32(mark(X1),X2))
p96: active#(U95(tt(),V2)) -> mark#(U96(isNatList(V2)))
p97: mark#(U32(X1,X2)) -> mark#(X1)
p98: active#(isNat(length(V1))) -> mark#(U11(isNatIListKind(V1),V1))
p99: mark#(U33(X)) -> mark#(X)
p100: mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3))
p101: active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1))
p102: mark#(U41(X1,X2,X3)) -> mark#(X1)
p103: mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3))
p104: active#(isNatIList(V)) -> mark#(U31(isNatIListKind(V),V))
p105: mark#(U42(X1,X2,X3)) -> mark#(X1)
p106: mark#(U43(X1,X2,X3)) -> active#(U43(mark(X1),X2,X3))
p107: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNatKind(V1),V1,V2))
p108: mark#(U43(X1,X2,X3)) -> mark#(X1)
p109: mark#(U44(X1,X2,X3)) -> active#(U44(mark(X1),X2,X3))
p110: active#(isNatIListKind(cons(V1,V2))) -> mark#(U51(isNatKind(V1),V2))
p111: mark#(U44(X1,X2,X3)) -> mark#(X1)
p112: mark#(U45(X1,X2)) -> active#(U45(mark(X1),X2))
p113: active#(isNatIListKind(take(V1,V2))) -> mark#(U61(isNatKind(V1),V2))
p114: mark#(U45(X1,X2)) -> mark#(X1)
p115: mark#(U46(X)) -> active#(U46(mark(X)))
p116: active#(isNatKind(length(V1))) -> mark#(U71(isNatIListKind(V1)))
p117: mark#(U46(X)) -> mark#(X)
p118: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p119: active#(isNatKind(s(V1))) -> mark#(U81(isNatKind(V1)))
p120: mark#(U51(X1,X2)) -> mark#(X1)
p121: mark#(U52(X)) -> active#(U52(mark(X)))
p122: active#(isNatList(cons(V1,V2))) -> mark#(U91(isNatKind(V1),V1,V2))
p123: mark#(U52(X)) -> mark#(X)
p124: mark#(U61(X1,X2)) -> active#(U61(mark(X1),X2))
p125: active#(isNatList(take(V1,V2))) -> mark#(U101(isNatKind(V1),V1,V2))
p126: mark#(U61(X1,X2)) -> mark#(X1)
p127: mark#(U62(X)) -> active#(U62(mark(X)))
p128: active#(length(cons(N,L))) -> mark#(U111(isNatList(L),L,N))
p129: mark#(U62(X)) -> mark#(X)
p130: active#(take(|0|(),IL)) -> mark#(U121(isNatIList(IL),IL))
p131: mark#(U71(X)) -> mark#(X)
p132: mark#(U81(X)) -> active#(U81(mark(X)))
p133: active#(take(s(M),cons(N,IL))) -> mark#(U131(isNatIList(IL),IL,M,N))
p134: mark#(U81(X)) -> mark#(X)
p135: mark#(U91(X1,X2,X3)) -> active#(U91(mark(X1),X2,X3))
p136: mark#(U91(X1,X2,X3)) -> mark#(X1)
p137: mark#(U92(X1,X2,X3)) -> active#(U92(mark(X1),X2,X3))
p138: mark#(U92(X1,X2,X3)) -> mark#(X1)
p139: mark#(U93(X1,X2,X3)) -> active#(U93(mark(X1),X2,X3))
p140: mark#(U93(X1,X2,X3)) -> mark#(X1)
p141: mark#(U94(X1,X2,X3)) -> active#(U94(mark(X1),X2,X3))
p142: mark#(U94(X1,X2,X3)) -> mark#(X1)
p143: mark#(U95(X1,X2)) -> active#(U95(mark(X1),X2))
p144: mark#(U95(X1,X2)) -> mark#(X1)
p145: mark#(U96(X)) -> mark#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19, p20, p21, p22, p23, p24, p25, p26, p27, p28, p29, p30, p31, p32, p33, p34, p35, p36, p37, p38, p39, p40, p41, p42, p43, p44, p45, p46, p47, p48, p49, p50, p51, p52, p53, p54, p55, p56, p57, p58, p59, p60, p61, p62, p63, p64, p65, p66, p67, p68, p69, p70, p71, p72, p73, p74, p75, p76, p77, p78, p79, p80, p81, p82, p83, p84, p85, p86, p87, p88, p89, p90, p91, p92, p93, p94, p95, p96, p97, p98, p99, p100, p101, p102, p103, p104, p105, p106, p107, p108, p109, p110, p111, p112, p113, p114, p115, p116, p117, p118, p119, p120, p121, p122, p123, p124, p125, p126, p127, p128, p129, p130, p131, p132, p133, p134, p135, p136, p137, p138, p139, p140, p141, p142, p143, p144, p145}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: active#(zeros()) -> mark#(cons(|0|(),zeros()))
p2: mark#(cons(X1,X2)) -> mark#(X1)
p3: mark#(U96(X)) -> mark#(X)
p4: mark#(U95(X1,X2)) -> mark#(X1)
p5: mark#(U95(X1,X2)) -> active#(U95(mark(X1),X2))
p6: active#(take(s(M),cons(N,IL))) -> mark#(U131(isNatIList(IL),IL,M,N))
p7: mark#(U94(X1,X2,X3)) -> mark#(X1)
p8: mark#(U94(X1,X2,X3)) -> active#(U94(mark(X1),X2,X3))
p9: active#(take(|0|(),IL)) -> mark#(U121(isNatIList(IL),IL))
p10: mark#(U93(X1,X2,X3)) -> mark#(X1)
p11: mark#(U93(X1,X2,X3)) -> active#(U93(mark(X1),X2,X3))
p12: active#(length(cons(N,L))) -> mark#(U111(isNatList(L),L,N))
p13: mark#(U92(X1,X2,X3)) -> mark#(X1)
p14: mark#(U92(X1,X2,X3)) -> active#(U92(mark(X1),X2,X3))
p15: active#(isNatList(take(V1,V2))) -> mark#(U101(isNatKind(V1),V1,V2))
p16: mark#(U91(X1,X2,X3)) -> mark#(X1)
p17: mark#(U91(X1,X2,X3)) -> active#(U91(mark(X1),X2,X3))
p18: active#(isNatList(cons(V1,V2))) -> mark#(U91(isNatKind(V1),V1,V2))
p19: mark#(U81(X)) -> mark#(X)
p20: mark#(U81(X)) -> active#(U81(mark(X)))
p21: active#(isNatKind(s(V1))) -> mark#(U81(isNatKind(V1)))
p22: mark#(U71(X)) -> mark#(X)
p23: mark#(U62(X)) -> mark#(X)
p24: mark#(U62(X)) -> active#(U62(mark(X)))
p25: active#(isNatKind(length(V1))) -> mark#(U71(isNatIListKind(V1)))
p26: mark#(U61(X1,X2)) -> mark#(X1)
p27: mark#(U61(X1,X2)) -> active#(U61(mark(X1),X2))
p28: active#(isNatIListKind(take(V1,V2))) -> mark#(U61(isNatKind(V1),V2))
p29: mark#(U52(X)) -> mark#(X)
p30: mark#(U52(X)) -> active#(U52(mark(X)))
p31: active#(isNatIListKind(cons(V1,V2))) -> mark#(U51(isNatKind(V1),V2))
p32: mark#(U51(X1,X2)) -> mark#(X1)
p33: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p34: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNatKind(V1),V1,V2))
p35: mark#(U46(X)) -> mark#(X)
p36: mark#(U46(X)) -> active#(U46(mark(X)))
p37: active#(isNatIList(V)) -> mark#(U31(isNatIListKind(V),V))
p38: mark#(U45(X1,X2)) -> mark#(X1)
p39: mark#(U45(X1,X2)) -> active#(U45(mark(X1),X2))
p40: active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1))
p41: mark#(U44(X1,X2,X3)) -> mark#(X1)
p42: mark#(U44(X1,X2,X3)) -> active#(U44(mark(X1),X2,X3))
p43: active#(isNat(length(V1))) -> mark#(U11(isNatIListKind(V1),V1))
p44: mark#(U43(X1,X2,X3)) -> mark#(X1)
p45: mark#(U43(X1,X2,X3)) -> active#(U43(mark(X1),X2,X3))
p46: active#(U95(tt(),V2)) -> mark#(U96(isNatList(V2)))
p47: mark#(U42(X1,X2,X3)) -> mark#(X1)
p48: mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3))
p49: active#(U94(tt(),V1,V2)) -> mark#(U95(isNat(V1),V2))
p50: mark#(U41(X1,X2,X3)) -> mark#(X1)
p51: mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3))
p52: active#(U93(tt(),V1,V2)) -> mark#(U94(isNatIListKind(V2),V1,V2))
p53: mark#(U33(X)) -> mark#(X)
p54: mark#(U32(X1,X2)) -> mark#(X1)
p55: mark#(U32(X1,X2)) -> active#(U32(mark(X1),X2))
p56: active#(U92(tt(),V1,V2)) -> mark#(U93(isNatIListKind(V2),V1,V2))
p57: mark#(U31(X1,X2)) -> mark#(X1)
p58: mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2))
p59: active#(U91(tt(),V1,V2)) -> mark#(U92(isNatKind(V1),V1,V2))
p60: mark#(U23(X)) -> mark#(X)
p61: mark#(U22(X1,X2)) -> mark#(X1)
p62: mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2))
p63: active#(U61(tt(),V2)) -> mark#(U62(isNatIListKind(V2)))
p64: mark#(U21(X1,X2)) -> mark#(X1)
p65: mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2))
p66: active#(U51(tt(),V2)) -> mark#(U52(isNatIListKind(V2)))
p67: mark#(take(X1,X2)) -> mark#(X2)
p68: mark#(take(X1,X2)) -> mark#(X1)
p69: mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2)))
p70: active#(U45(tt(),V2)) -> mark#(U46(isNatIList(V2)))
p71: mark#(U136(X1,X2,X3,X4)) -> mark#(X1)
p72: mark#(U136(X1,X2,X3,X4)) -> active#(U136(mark(X1),X2,X3,X4))
p73: active#(U44(tt(),V1,V2)) -> mark#(U45(isNat(V1),V2))
p74: mark#(U135(X1,X2,X3,X4)) -> mark#(X1)
p75: mark#(U135(X1,X2,X3,X4)) -> active#(U135(mark(X1),X2,X3,X4))
p76: active#(U43(tt(),V1,V2)) -> mark#(U44(isNatIListKind(V2),V1,V2))
p77: mark#(U134(X1,X2,X3,X4)) -> mark#(X1)
p78: mark#(U134(X1,X2,X3,X4)) -> active#(U134(mark(X1),X2,X3,X4))
p79: active#(U42(tt(),V1,V2)) -> mark#(U43(isNatIListKind(V2),V1,V2))
p80: mark#(U133(X1,X2,X3,X4)) -> mark#(X1)
p81: mark#(U133(X1,X2,X3,X4)) -> active#(U133(mark(X1),X2,X3,X4))
p82: active#(U41(tt(),V1,V2)) -> mark#(U42(isNatKind(V1),V1,V2))
p83: mark#(U132(X1,X2,X3,X4)) -> mark#(X1)
p84: mark#(U132(X1,X2,X3,X4)) -> active#(U132(mark(X1),X2,X3,X4))
p85: active#(U32(tt(),V)) -> mark#(U33(isNatList(V)))
p86: mark#(U131(X1,X2,X3,X4)) -> mark#(X1)
p87: mark#(U131(X1,X2,X3,X4)) -> active#(U131(mark(X1),X2,X3,X4))
p88: active#(U31(tt(),V)) -> mark#(U32(isNatIListKind(V),V))
p89: mark#(U122(X)) -> mark#(X)
p90: mark#(U121(X1,X2)) -> mark#(X1)
p91: mark#(U121(X1,X2)) -> active#(U121(mark(X1),X2))
p92: active#(U22(tt(),V1)) -> mark#(U23(isNat(V1)))
p93: mark#(isNatList(X)) -> active#(isNatList(X))
p94: active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1))
p95: mark#(U13(X)) -> mark#(X)
p96: mark#(length(X)) -> mark#(X)
p97: mark#(length(X)) -> active#(length(mark(X)))
p98: active#(U136(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL)))
p99: mark#(s(X)) -> mark#(X)
p100: mark#(s(X)) -> active#(s(mark(X)))
p101: active#(U135(tt(),IL,M,N)) -> mark#(U136(isNatKind(N),IL,M,N))
p102: mark#(U114(X1,X2)) -> mark#(X1)
p103: mark#(U114(X1,X2)) -> active#(U114(mark(X1),X2))
p104: active#(U134(tt(),IL,M,N)) -> mark#(U135(isNat(N),IL,M,N))
p105: mark#(U113(X1,X2,X3)) -> mark#(X1)
p106: mark#(U113(X1,X2,X3)) -> active#(U113(mark(X1),X2,X3))
p107: active#(U133(tt(),IL,M,N)) -> mark#(U134(isNatKind(M),IL,M,N))
p108: mark#(U112(X1,X2,X3)) -> mark#(X1)
p109: mark#(U112(X1,X2,X3)) -> active#(U112(mark(X1),X2,X3))
p110: active#(U132(tt(),IL,M,N)) -> mark#(U133(isNat(M),IL,M,N))
p111: mark#(U111(X1,X2,X3)) -> mark#(X1)
p112: mark#(U111(X1,X2,X3)) -> active#(U111(mark(X1),X2,X3))
p113: active#(U131(tt(),IL,M,N)) -> mark#(U132(isNatIListKind(IL),IL,M,N))
p114: mark#(U12(X1,X2)) -> mark#(X1)
p115: mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2))
p116: active#(U121(tt(),IL)) -> mark#(U122(isNatIListKind(IL)))
p117: mark#(U11(X1,X2)) -> mark#(X1)
p118: mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2))
p119: active#(U12(tt(),V1)) -> mark#(U13(isNatList(V1)))
p120: mark#(isNatIList(X)) -> active#(isNatIList(X))
p121: active#(U114(tt(),L)) -> mark#(s(length(L)))
p122: mark#(U106(X)) -> mark#(X)
p123: mark#(isNat(X)) -> active#(isNat(X))
p124: active#(U113(tt(),L,N)) -> mark#(U114(isNatKind(N),L))
p125: mark#(U105(X1,X2)) -> mark#(X1)
p126: mark#(U105(X1,X2)) -> active#(U105(mark(X1),X2))
p127: active#(U112(tt(),L,N)) -> mark#(U113(isNat(N),L,N))
p128: mark#(U104(X1,X2,X3)) -> mark#(X1)
p129: mark#(U104(X1,X2,X3)) -> active#(U104(mark(X1),X2,X3))
p130: active#(U111(tt(),L,N)) -> mark#(U112(isNatIListKind(L),L,N))
p131: mark#(isNatIListKind(X)) -> active#(isNatIListKind(X))
p132: active#(U11(tt(),V1)) -> mark#(U12(isNatIListKind(V1),V1))
p133: mark#(U103(X1,X2,X3)) -> mark#(X1)
p134: mark#(U103(X1,X2,X3)) -> active#(U103(mark(X1),X2,X3))
p135: active#(U105(tt(),V2)) -> mark#(U106(isNatIList(V2)))
p136: mark#(isNatKind(X)) -> active#(isNatKind(X))
p137: active#(U104(tt(),V1,V2)) -> mark#(U105(isNat(V1),V2))
p138: mark#(U102(X1,X2,X3)) -> mark#(X1)
p139: mark#(U102(X1,X2,X3)) -> active#(U102(mark(X1),X2,X3))
p140: active#(U103(tt(),V1,V2)) -> mark#(U104(isNatIListKind(V2),V1,V2))
p141: mark#(U101(X1,X2,X3)) -> mark#(X1)
p142: mark#(U101(X1,X2,X3)) -> active#(U101(mark(X1),X2,X3))
p143: active#(U102(tt(),V1,V2)) -> mark#(U103(isNatIListKind(V2),V1,V2))
p144: mark#(zeros()) -> active#(zeros())
p145: active#(U101(tt(),V1,V2)) -> mark#(U102(isNatKind(V1),V1,V2))

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41, r42, r43, r44, r45, r46, r47, r48, r49, r50, r51, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76, r77, r78, r79, r80, r81, r82, r83, r84, r85, r86, r87, r88, r89, r90, r91, r92, r93, r94, r95, r96, r97, r98, r99, r100, r101, r102, r103, r104, r105, r106, r107, r108, r109, r110, r111, r112, r113, r114, r115, r116, r117, r118, r119, r120, r121, r122, r123, r124, r125, r126, r127, r128, r129, r130, r131, r132, r133, r134, r135, r136, r137, r138, r139, r140, r141, r142, r143, r144, r145, r146, r147, r148, r149, r150, r151, r152, r153, r154, r155, r156, r157, r158, r159, r160, r161, r162, r163, r164, r165, r166, r167, r168, r169, r170, r171, r172, r173, r174, r175, r176, r177, r178, r179, r180, r181, r182, r183, r184, r185, r186, r187, r188, r189, r190, r191, r192, r193, r194, r195, r196, r197, r198, r199, r200, r201, r202, r203, r204, r205, r206, r207, r208, r209, r210, r211, r212, r213, r214, r215, r216, r217, r218, r219, r220, r221, r222, r223, r224, r225, r226, r227, r228, r229, r230, r231, r232, r233, r234, r235, r236, r237, r238, r239, r240, r241, r242, r243, r244, r245, r246, r247, r248, r249, r250, r251, r252, r253, r254, r255, r256, r257, r258, r259, r260, r261, r262, r263, r264, r265, r266, r267, r268, r269, r270, r271, r272, r273, r274, r275, r276, r277, r278, r279, r280, r281, r282, r283, r284, r285, r286, r287, r288, r289, r290, r291, r292, r293, r294, r295, r296, r297, r298, r299, r300, r301, r302, r303, r304, r305, r306, r307, r308, r309, r310, r311, r312, r313, r314, r315, r316, r317, r318, r319, r320, r321, r322, r323, r324, r325, r326, r327, r328, r329, r330, r331, r332, r333, r334, r335, r336, r337, r338, r339, r340, r341, r342, r343, r344, r345, r346, r347, r348, r349, r350, r351, r352, r353, r354, r355, r356, r357, r358

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    active#_A(x1) = x1 + 98
    zeros_A = 214
    mark#_A(x1) = 312
    cons_A(x1,x2) = 213
    |0|_A = 212
    U96_A(x1) = 214
    U95_A(x1,x2) = 214
    mark_A(x1) = max{59, x1 - 1}
    take_A(x1,x2) = 214
    s_A(x1) = 214
    U131_A(x1,x2,x3,x4) = 214
    isNatIList_A(x1) = 214
    U94_A(x1,x2,x3) = 214
    U121_A(x1,x2) = 214
    U93_A(x1,x2,x3) = 214
    length_A(x1) = 214
    U111_A(x1,x2,x3) = 214
    isNatList_A(x1) = 214
    U92_A(x1,x2,x3) = 214
    U101_A(x1,x2,x3) = 214
    isNatKind_A(x1) = 214
    U91_A(x1,x2,x3) = 214
    U81_A(x1) = 214
    U71_A(x1) = 212
    U62_A(x1) = 214
    isNatIListKind_A(x1) = 214
    U61_A(x1,x2) = 214
    U52_A(x1) = 213
    U51_A(x1,x2) = 214
    U41_A(x1,x2,x3) = 214
    U46_A(x1) = 214
    U31_A(x1,x2) = 214
    U45_A(x1,x2) = 214
    isNat_A(x1) = 214
    U21_A(x1,x2) = 214
    U44_A(x1,x2,x3) = 214
    U11_A(x1,x2) = 214
    U43_A(x1,x2,x3) = 214
    tt_A = 63
    U42_A(x1,x2,x3) = 214
    U33_A(x1) = 214
    U32_A(x1,x2) = 214
    U23_A(x1) = 214
    U22_A(x1,x2) = 214
    U136_A(x1,x2,x3,x4) = 214
    U135_A(x1,x2,x3,x4) = 214
    U134_A(x1,x2,x3,x4) = 214
    U133_A(x1,x2,x3,x4) = 214
    U132_A(x1,x2,x3,x4) = 214
    U122_A(x1) = 63
    U13_A(x1) = 214
    U114_A(x1,x2) = 214
    U113_A(x1,x2,x3) = 214
    U112_A(x1,x2,x3) = 214
    U12_A(x1,x2) = 214
    U106_A(x1) = 63
    U105_A(x1,x2) = 214
    U104_A(x1,x2,x3) = 214
    U103_A(x1,x2,x3) = 214
    U102_A(x1,x2,x3) = 214
    active_A(x1) = max{61, x1 - 1}
    nil_A = 63

The next rules are strictly ordered:

  p30

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: active#(zeros()) -> mark#(cons(|0|(),zeros()))
p2: mark#(cons(X1,X2)) -> mark#(X1)
p3: mark#(U96(X)) -> mark#(X)
p4: mark#(U95(X1,X2)) -> mark#(X1)
p5: mark#(U95(X1,X2)) -> active#(U95(mark(X1),X2))
p6: active#(take(s(M),cons(N,IL))) -> mark#(U131(isNatIList(IL),IL,M,N))
p7: mark#(U94(X1,X2,X3)) -> mark#(X1)
p8: mark#(U94(X1,X2,X3)) -> active#(U94(mark(X1),X2,X3))
p9: active#(take(|0|(),IL)) -> mark#(U121(isNatIList(IL),IL))
p10: mark#(U93(X1,X2,X3)) -> mark#(X1)
p11: mark#(U93(X1,X2,X3)) -> active#(U93(mark(X1),X2,X3))
p12: active#(length(cons(N,L))) -> mark#(U111(isNatList(L),L,N))
p13: mark#(U92(X1,X2,X3)) -> mark#(X1)
p14: mark#(U92(X1,X2,X3)) -> active#(U92(mark(X1),X2,X3))
p15: active#(isNatList(take(V1,V2))) -> mark#(U101(isNatKind(V1),V1,V2))
p16: mark#(U91(X1,X2,X3)) -> mark#(X1)
p17: mark#(U91(X1,X2,X3)) -> active#(U91(mark(X1),X2,X3))
p18: active#(isNatList(cons(V1,V2))) -> mark#(U91(isNatKind(V1),V1,V2))
p19: mark#(U81(X)) -> mark#(X)
p20: mark#(U81(X)) -> active#(U81(mark(X)))
p21: active#(isNatKind(s(V1))) -> mark#(U81(isNatKind(V1)))
p22: mark#(U71(X)) -> mark#(X)
p23: mark#(U62(X)) -> mark#(X)
p24: mark#(U62(X)) -> active#(U62(mark(X)))
p25: active#(isNatKind(length(V1))) -> mark#(U71(isNatIListKind(V1)))
p26: mark#(U61(X1,X2)) -> mark#(X1)
p27: mark#(U61(X1,X2)) -> active#(U61(mark(X1),X2))
p28: active#(isNatIListKind(take(V1,V2))) -> mark#(U61(isNatKind(V1),V2))
p29: mark#(U52(X)) -> mark#(X)
p30: active#(isNatIListKind(cons(V1,V2))) -> mark#(U51(isNatKind(V1),V2))
p31: mark#(U51(X1,X2)) -> mark#(X1)
p32: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p33: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNatKind(V1),V1,V2))
p34: mark#(U46(X)) -> mark#(X)
p35: mark#(U46(X)) -> active#(U46(mark(X)))
p36: active#(isNatIList(V)) -> mark#(U31(isNatIListKind(V),V))
p37: mark#(U45(X1,X2)) -> mark#(X1)
p38: mark#(U45(X1,X2)) -> active#(U45(mark(X1),X2))
p39: active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1))
p40: mark#(U44(X1,X2,X3)) -> mark#(X1)
p41: mark#(U44(X1,X2,X3)) -> active#(U44(mark(X1),X2,X3))
p42: active#(isNat(length(V1))) -> mark#(U11(isNatIListKind(V1),V1))
p43: mark#(U43(X1,X2,X3)) -> mark#(X1)
p44: mark#(U43(X1,X2,X3)) -> active#(U43(mark(X1),X2,X3))
p45: active#(U95(tt(),V2)) -> mark#(U96(isNatList(V2)))
p46: mark#(U42(X1,X2,X3)) -> mark#(X1)
p47: mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3))
p48: active#(U94(tt(),V1,V2)) -> mark#(U95(isNat(V1),V2))
p49: mark#(U41(X1,X2,X3)) -> mark#(X1)
p50: mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3))
p51: active#(U93(tt(),V1,V2)) -> mark#(U94(isNatIListKind(V2),V1,V2))
p52: mark#(U33(X)) -> mark#(X)
p53: mark#(U32(X1,X2)) -> mark#(X1)
p54: mark#(U32(X1,X2)) -> active#(U32(mark(X1),X2))
p55: active#(U92(tt(),V1,V2)) -> mark#(U93(isNatIListKind(V2),V1,V2))
p56: mark#(U31(X1,X2)) -> mark#(X1)
p57: mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2))
p58: active#(U91(tt(),V1,V2)) -> mark#(U92(isNatKind(V1),V1,V2))
p59: mark#(U23(X)) -> mark#(X)
p60: mark#(U22(X1,X2)) -> mark#(X1)
p61: mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2))
p62: active#(U61(tt(),V2)) -> mark#(U62(isNatIListKind(V2)))
p63: mark#(U21(X1,X2)) -> mark#(X1)
p64: mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2))
p65: active#(U51(tt(),V2)) -> mark#(U52(isNatIListKind(V2)))
p66: mark#(take(X1,X2)) -> mark#(X2)
p67: mark#(take(X1,X2)) -> mark#(X1)
p68: mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2)))
p69: active#(U45(tt(),V2)) -> mark#(U46(isNatIList(V2)))
p70: mark#(U136(X1,X2,X3,X4)) -> mark#(X1)
p71: mark#(U136(X1,X2,X3,X4)) -> active#(U136(mark(X1),X2,X3,X4))
p72: active#(U44(tt(),V1,V2)) -> mark#(U45(isNat(V1),V2))
p73: mark#(U135(X1,X2,X3,X4)) -> mark#(X1)
p74: mark#(U135(X1,X2,X3,X4)) -> active#(U135(mark(X1),X2,X3,X4))
p75: active#(U43(tt(),V1,V2)) -> mark#(U44(isNatIListKind(V2),V1,V2))
p76: mark#(U134(X1,X2,X3,X4)) -> mark#(X1)
p77: mark#(U134(X1,X2,X3,X4)) -> active#(U134(mark(X1),X2,X3,X4))
p78: active#(U42(tt(),V1,V2)) -> mark#(U43(isNatIListKind(V2),V1,V2))
p79: mark#(U133(X1,X2,X3,X4)) -> mark#(X1)
p80: mark#(U133(X1,X2,X3,X4)) -> active#(U133(mark(X1),X2,X3,X4))
p81: active#(U41(tt(),V1,V2)) -> mark#(U42(isNatKind(V1),V1,V2))
p82: mark#(U132(X1,X2,X3,X4)) -> mark#(X1)
p83: mark#(U132(X1,X2,X3,X4)) -> active#(U132(mark(X1),X2,X3,X4))
p84: active#(U32(tt(),V)) -> mark#(U33(isNatList(V)))
p85: mark#(U131(X1,X2,X3,X4)) -> mark#(X1)
p86: mark#(U131(X1,X2,X3,X4)) -> active#(U131(mark(X1),X2,X3,X4))
p87: active#(U31(tt(),V)) -> mark#(U32(isNatIListKind(V),V))
p88: mark#(U122(X)) -> mark#(X)
p89: mark#(U121(X1,X2)) -> mark#(X1)
p90: mark#(U121(X1,X2)) -> active#(U121(mark(X1),X2))
p91: active#(U22(tt(),V1)) -> mark#(U23(isNat(V1)))
p92: mark#(isNatList(X)) -> active#(isNatList(X))
p93: active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1))
p94: mark#(U13(X)) -> mark#(X)
p95: mark#(length(X)) -> mark#(X)
p96: mark#(length(X)) -> active#(length(mark(X)))
p97: active#(U136(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL)))
p98: mark#(s(X)) -> mark#(X)
p99: mark#(s(X)) -> active#(s(mark(X)))
p100: active#(U135(tt(),IL,M,N)) -> mark#(U136(isNatKind(N),IL,M,N))
p101: mark#(U114(X1,X2)) -> mark#(X1)
p102: mark#(U114(X1,X2)) -> active#(U114(mark(X1),X2))
p103: active#(U134(tt(),IL,M,N)) -> mark#(U135(isNat(N),IL,M,N))
p104: mark#(U113(X1,X2,X3)) -> mark#(X1)
p105: mark#(U113(X1,X2,X3)) -> active#(U113(mark(X1),X2,X3))
p106: active#(U133(tt(),IL,M,N)) -> mark#(U134(isNatKind(M),IL,M,N))
p107: mark#(U112(X1,X2,X3)) -> mark#(X1)
p108: mark#(U112(X1,X2,X3)) -> active#(U112(mark(X1),X2,X3))
p109: active#(U132(tt(),IL,M,N)) -> mark#(U133(isNat(M),IL,M,N))
p110: mark#(U111(X1,X2,X3)) -> mark#(X1)
p111: mark#(U111(X1,X2,X3)) -> active#(U111(mark(X1),X2,X3))
p112: active#(U131(tt(),IL,M,N)) -> mark#(U132(isNatIListKind(IL),IL,M,N))
p113: mark#(U12(X1,X2)) -> mark#(X1)
p114: mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2))
p115: active#(U121(tt(),IL)) -> mark#(U122(isNatIListKind(IL)))
p116: mark#(U11(X1,X2)) -> mark#(X1)
p117: mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2))
p118: active#(U12(tt(),V1)) -> mark#(U13(isNatList(V1)))
p119: mark#(isNatIList(X)) -> active#(isNatIList(X))
p120: active#(U114(tt(),L)) -> mark#(s(length(L)))
p121: mark#(U106(X)) -> mark#(X)
p122: mark#(isNat(X)) -> active#(isNat(X))
p123: active#(U113(tt(),L,N)) -> mark#(U114(isNatKind(N),L))
p124: mark#(U105(X1,X2)) -> mark#(X1)
p125: mark#(U105(X1,X2)) -> active#(U105(mark(X1),X2))
p126: active#(U112(tt(),L,N)) -> mark#(U113(isNat(N),L,N))
p127: mark#(U104(X1,X2,X3)) -> mark#(X1)
p128: mark#(U104(X1,X2,X3)) -> active#(U104(mark(X1),X2,X3))
p129: active#(U111(tt(),L,N)) -> mark#(U112(isNatIListKind(L),L,N))
p130: mark#(isNatIListKind(X)) -> active#(isNatIListKind(X))
p131: active#(U11(tt(),V1)) -> mark#(U12(isNatIListKind(V1),V1))
p132: mark#(U103(X1,X2,X3)) -> mark#(X1)
p133: mark#(U103(X1,X2,X3)) -> active#(U103(mark(X1),X2,X3))
p134: active#(U105(tt(),V2)) -> mark#(U106(isNatIList(V2)))
p135: mark#(isNatKind(X)) -> active#(isNatKind(X))
p136: active#(U104(tt(),V1,V2)) -> mark#(U105(isNat(V1),V2))
p137: mark#(U102(X1,X2,X3)) -> mark#(X1)
p138: mark#(U102(X1,X2,X3)) -> active#(U102(mark(X1),X2,X3))
p139: active#(U103(tt(),V1,V2)) -> mark#(U104(isNatIListKind(V2),V1,V2))
p140: mark#(U101(X1,X2,X3)) -> mark#(X1)
p141: mark#(U101(X1,X2,X3)) -> active#(U101(mark(X1),X2,X3))
p142: active#(U102(tt(),V1,V2)) -> mark#(U103(isNatIListKind(V2),V1,V2))
p143: mark#(zeros()) -> active#(zeros())
p144: active#(U101(tt(),V1,V2)) -> mark#(U102(isNatKind(V1),V1,V2))

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19, p20, p21, p22, p23, p24, p25, p26, p27, p28, p29, p30, p31, p32, p33, p34, p35, p36, p37, p38, p39, p40, p41, p42, p43, p44, p45, p46, p47, p48, p49, p50, p51, p52, p53, p54, p55, p56, p57, p58, p59, p60, p61, p62, p63, p64, p65, p66, p67, p68, p69, p70, p71, p72, p73, p74, p75, p76, p77, p78, p79, p80, p81, p82, p83, p84, p85, p86, p87, p88, p89, p90, p91, p92, p93, p94, p95, p96, p97, p98, p99, p100, p101, p102, p103, p104, p105, p106, p107, p108, p109, p110, p111, p112, p113, p114, p115, p116, p117, p118, p119, p120, p121, p122, p123, p124, p125, p126, p127, p128, p129, p130, p131, p132, p133, p134, p135, p136, p137, p138, p139, p140, p141, p142, p143, p144}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: active#(zeros()) -> mark#(cons(|0|(),zeros()))
p2: mark#(cons(X1,X2)) -> mark#(X1)
p3: mark#(zeros()) -> active#(zeros())
p4: mark#(U101(X1,X2,X3)) -> active#(U101(mark(X1),X2,X3))
p5: active#(U101(tt(),V1,V2)) -> mark#(U102(isNatKind(V1),V1,V2))
p6: mark#(U101(X1,X2,X3)) -> mark#(X1)
p7: mark#(U102(X1,X2,X3)) -> active#(U102(mark(X1),X2,X3))
p8: active#(U102(tt(),V1,V2)) -> mark#(U103(isNatIListKind(V2),V1,V2))
p9: mark#(U102(X1,X2,X3)) -> mark#(X1)
p10: mark#(isNatKind(X)) -> active#(isNatKind(X))
p11: active#(U103(tt(),V1,V2)) -> mark#(U104(isNatIListKind(V2),V1,V2))
p12: mark#(U103(X1,X2,X3)) -> active#(U103(mark(X1),X2,X3))
p13: active#(U104(tt(),V1,V2)) -> mark#(U105(isNat(V1),V2))
p14: mark#(U103(X1,X2,X3)) -> mark#(X1)
p15: mark#(isNatIListKind(X)) -> active#(isNatIListKind(X))
p16: active#(U105(tt(),V2)) -> mark#(U106(isNatIList(V2)))
p17: mark#(U104(X1,X2,X3)) -> active#(U104(mark(X1),X2,X3))
p18: active#(U11(tt(),V1)) -> mark#(U12(isNatIListKind(V1),V1))
p19: mark#(U104(X1,X2,X3)) -> mark#(X1)
p20: mark#(U105(X1,X2)) -> active#(U105(mark(X1),X2))
p21: active#(U111(tt(),L,N)) -> mark#(U112(isNatIListKind(L),L,N))
p22: mark#(U105(X1,X2)) -> mark#(X1)
p23: mark#(isNat(X)) -> active#(isNat(X))
p24: active#(U112(tt(),L,N)) -> mark#(U113(isNat(N),L,N))
p25: mark#(U106(X)) -> mark#(X)
p26: mark#(isNatIList(X)) -> active#(isNatIList(X))
p27: active#(U113(tt(),L,N)) -> mark#(U114(isNatKind(N),L))
p28: mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2))
p29: active#(U114(tt(),L)) -> mark#(s(length(L)))
p30: mark#(U11(X1,X2)) -> mark#(X1)
p31: mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2))
p32: active#(U12(tt(),V1)) -> mark#(U13(isNatList(V1)))
p33: mark#(U12(X1,X2)) -> mark#(X1)
p34: mark#(U111(X1,X2,X3)) -> active#(U111(mark(X1),X2,X3))
p35: active#(U121(tt(),IL)) -> mark#(U122(isNatIListKind(IL)))
p36: mark#(U111(X1,X2,X3)) -> mark#(X1)
p37: mark#(U112(X1,X2,X3)) -> active#(U112(mark(X1),X2,X3))
p38: active#(U131(tt(),IL,M,N)) -> mark#(U132(isNatIListKind(IL),IL,M,N))
p39: mark#(U112(X1,X2,X3)) -> mark#(X1)
p40: mark#(U113(X1,X2,X3)) -> active#(U113(mark(X1),X2,X3))
p41: active#(U132(tt(),IL,M,N)) -> mark#(U133(isNat(M),IL,M,N))
p42: mark#(U113(X1,X2,X3)) -> mark#(X1)
p43: mark#(U114(X1,X2)) -> active#(U114(mark(X1),X2))
p44: active#(U133(tt(),IL,M,N)) -> mark#(U134(isNatKind(M),IL,M,N))
p45: mark#(U114(X1,X2)) -> mark#(X1)
p46: mark#(s(X)) -> active#(s(mark(X)))
p47: active#(U134(tt(),IL,M,N)) -> mark#(U135(isNat(N),IL,M,N))
p48: mark#(s(X)) -> mark#(X)
p49: mark#(length(X)) -> active#(length(mark(X)))
p50: active#(U135(tt(),IL,M,N)) -> mark#(U136(isNatKind(N),IL,M,N))
p51: mark#(length(X)) -> mark#(X)
p52: mark#(U13(X)) -> mark#(X)
p53: mark#(isNatList(X)) -> active#(isNatList(X))
p54: active#(U136(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL)))
p55: mark#(U121(X1,X2)) -> active#(U121(mark(X1),X2))
p56: active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1))
p57: mark#(U121(X1,X2)) -> mark#(X1)
p58: mark#(U122(X)) -> mark#(X)
p59: mark#(U131(X1,X2,X3,X4)) -> active#(U131(mark(X1),X2,X3,X4))
p60: active#(U22(tt(),V1)) -> mark#(U23(isNat(V1)))
p61: mark#(U131(X1,X2,X3,X4)) -> mark#(X1)
p62: mark#(U132(X1,X2,X3,X4)) -> active#(U132(mark(X1),X2,X3,X4))
p63: active#(U31(tt(),V)) -> mark#(U32(isNatIListKind(V),V))
p64: mark#(U132(X1,X2,X3,X4)) -> mark#(X1)
p65: mark#(U133(X1,X2,X3,X4)) -> active#(U133(mark(X1),X2,X3,X4))
p66: active#(U32(tt(),V)) -> mark#(U33(isNatList(V)))
p67: mark#(U133(X1,X2,X3,X4)) -> mark#(X1)
p68: mark#(U134(X1,X2,X3,X4)) -> active#(U134(mark(X1),X2,X3,X4))
p69: active#(U41(tt(),V1,V2)) -> mark#(U42(isNatKind(V1),V1,V2))
p70: mark#(U134(X1,X2,X3,X4)) -> mark#(X1)
p71: mark#(U135(X1,X2,X3,X4)) -> active#(U135(mark(X1),X2,X3,X4))
p72: active#(U42(tt(),V1,V2)) -> mark#(U43(isNatIListKind(V2),V1,V2))
p73: mark#(U135(X1,X2,X3,X4)) -> mark#(X1)
p74: mark#(U136(X1,X2,X3,X4)) -> active#(U136(mark(X1),X2,X3,X4))
p75: active#(U43(tt(),V1,V2)) -> mark#(U44(isNatIListKind(V2),V1,V2))
p76: mark#(U136(X1,X2,X3,X4)) -> mark#(X1)
p77: mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2)))
p78: active#(U44(tt(),V1,V2)) -> mark#(U45(isNat(V1),V2))
p79: mark#(take(X1,X2)) -> mark#(X1)
p80: mark#(take(X1,X2)) -> mark#(X2)
p81: mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2))
p82: active#(U45(tt(),V2)) -> mark#(U46(isNatIList(V2)))
p83: mark#(U21(X1,X2)) -> mark#(X1)
p84: mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2))
p85: active#(U51(tt(),V2)) -> mark#(U52(isNatIListKind(V2)))
p86: mark#(U22(X1,X2)) -> mark#(X1)
p87: mark#(U23(X)) -> mark#(X)
p88: mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2))
p89: active#(U61(tt(),V2)) -> mark#(U62(isNatIListKind(V2)))
p90: mark#(U31(X1,X2)) -> mark#(X1)
p91: mark#(U32(X1,X2)) -> active#(U32(mark(X1),X2))
p92: active#(U91(tt(),V1,V2)) -> mark#(U92(isNatKind(V1),V1,V2))
p93: mark#(U32(X1,X2)) -> mark#(X1)
p94: mark#(U33(X)) -> mark#(X)
p95: mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3))
p96: active#(U92(tt(),V1,V2)) -> mark#(U93(isNatIListKind(V2),V1,V2))
p97: mark#(U41(X1,X2,X3)) -> mark#(X1)
p98: mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3))
p99: active#(U93(tt(),V1,V2)) -> mark#(U94(isNatIListKind(V2),V1,V2))
p100: mark#(U42(X1,X2,X3)) -> mark#(X1)
p101: mark#(U43(X1,X2,X3)) -> active#(U43(mark(X1),X2,X3))
p102: active#(U94(tt(),V1,V2)) -> mark#(U95(isNat(V1),V2))
p103: mark#(U43(X1,X2,X3)) -> mark#(X1)
p104: mark#(U44(X1,X2,X3)) -> active#(U44(mark(X1),X2,X3))
p105: active#(U95(tt(),V2)) -> mark#(U96(isNatList(V2)))
p106: mark#(U44(X1,X2,X3)) -> mark#(X1)
p107: mark#(U45(X1,X2)) -> active#(U45(mark(X1),X2))
p108: active#(isNat(length(V1))) -> mark#(U11(isNatIListKind(V1),V1))
p109: mark#(U45(X1,X2)) -> mark#(X1)
p110: mark#(U46(X)) -> active#(U46(mark(X)))
p111: active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1))
p112: mark#(U46(X)) -> mark#(X)
p113: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p114: active#(isNatIList(V)) -> mark#(U31(isNatIListKind(V),V))
p115: mark#(U51(X1,X2)) -> mark#(X1)
p116: mark#(U52(X)) -> mark#(X)
p117: mark#(U61(X1,X2)) -> active#(U61(mark(X1),X2))
p118: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNatKind(V1),V1,V2))
p119: mark#(U61(X1,X2)) -> mark#(X1)
p120: mark#(U62(X)) -> active#(U62(mark(X)))
p121: active#(isNatIListKind(cons(V1,V2))) -> mark#(U51(isNatKind(V1),V2))
p122: mark#(U62(X)) -> mark#(X)
p123: mark#(U71(X)) -> mark#(X)
p124: mark#(U81(X)) -> active#(U81(mark(X)))
p125: active#(isNatIListKind(take(V1,V2))) -> mark#(U61(isNatKind(V1),V2))
p126: mark#(U81(X)) -> mark#(X)
p127: mark#(U91(X1,X2,X3)) -> active#(U91(mark(X1),X2,X3))
p128: active#(isNatKind(length(V1))) -> mark#(U71(isNatIListKind(V1)))
p129: mark#(U91(X1,X2,X3)) -> mark#(X1)
p130: mark#(U92(X1,X2,X3)) -> active#(U92(mark(X1),X2,X3))
p131: active#(isNatKind(s(V1))) -> mark#(U81(isNatKind(V1)))
p132: mark#(U92(X1,X2,X3)) -> mark#(X1)
p133: mark#(U93(X1,X2,X3)) -> active#(U93(mark(X1),X2,X3))
p134: active#(isNatList(cons(V1,V2))) -> mark#(U91(isNatKind(V1),V1,V2))
p135: mark#(U93(X1,X2,X3)) -> mark#(X1)
p136: mark#(U94(X1,X2,X3)) -> active#(U94(mark(X1),X2,X3))
p137: active#(isNatList(take(V1,V2))) -> mark#(U101(isNatKind(V1),V1,V2))
p138: mark#(U94(X1,X2,X3)) -> mark#(X1)
p139: mark#(U95(X1,X2)) -> active#(U95(mark(X1),X2))
p140: active#(length(cons(N,L))) -> mark#(U111(isNatList(L),L,N))
p141: mark#(U95(X1,X2)) -> mark#(X1)
p142: mark#(U96(X)) -> mark#(X)
p143: active#(take(|0|(),IL)) -> mark#(U121(isNatIList(IL),IL))
p144: active#(take(s(M),cons(N,IL))) -> mark#(U131(isNatIList(IL),IL,M,N))

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41, r42, r43, r44, r45, r46, r47, r48, r49, r50, r51, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76, r77, r78, r79, r80, r81, r82, r83, r84, r85, r86, r87, r88, r89, r90, r91, r92, r93, r94, r95, r96, r97, r98, r99, r100, r101, r102, r103, r104, r105, r106, r107, r108, r109, r110, r111, r112, r113, r114, r115, r116, r117, r118, r119, r120, r121, r122, r123, r124, r125, r126, r127, r128, r129, r130, r131, r132, r133, r134, r135, r136, r137, r138, r139, r140, r141, r142, r143, r144, r145, r146, r147, r148, r149, r150, r151, r152, r153, r154, r155, r156, r157, r158, r159, r160, r161, r162, r163, r164, r165, r166, r167, r168, r169, r170, r171, r172, r173, r174, r175, r176, r177, r178, r179, r180, r181, r182, r183, r184, r185, r186, r187, r188, r189, r190, r191, r192, r193, r194, r195, r196, r197, r198, r199, r200, r201, r202, r203, r204, r205, r206, r207, r208, r209, r210, r211, r212, r213, r214, r215, r216, r217, r218, r219, r220, r221, r222, r223, r224, r225, r226, r227, r228, r229, r230, r231, r232, r233, r234, r235, r236, r237, r238, r239, r240, r241, r242, r243, r244, r245, r246, r247, r248, r249, r250, r251, r252, r253, r254, r255, r256, r257, r258, r259, r260, r261, r262, r263, r264, r265, r266, r267, r268, r269, r270, r271, r272, r273, r274, r275, r276, r277, r278, r279, r280, r281, r282, r283, r284, r285, r286, r287, r288, r289, r290, r291, r292, r293, r294, r295, r296, r297, r298, r299, r300, r301, r302, r303, r304, r305, r306, r307, r308, r309, r310, r311, r312, r313, r314, r315, r316, r317, r318, r319, r320, r321, r322, r323, r324, r325, r326, r327, r328, r329, r330, r331, r332, r333, r334, r335, r336, r337, r338, r339, r340, r341, r342, r343, r344, r345, r346, r347, r348, r349, r350, r351, r352, r353, r354, r355, r356, r357, r358

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    active#_A(x1) = x1 + 7
    zeros_A = 383
    mark#_A(x1) = 390
    cons_A(x1,x2) = 255
    |0|_A = 258
    U101_A(x1,x2,x3) = 383
    mark_A(x1) = max{227, x1 - 1}
    tt_A = 255
    U102_A(x1,x2,x3) = 383
    isNatKind_A(x1) = 383
    U103_A(x1,x2,x3) = 383
    isNatIListKind_A(x1) = 383
    U104_A(x1,x2,x3) = 383
    U105_A(x1,x2) = 383
    isNat_A(x1) = 383
    U106_A(x1) = 381
    isNatIList_A(x1) = 383
    U11_A(x1,x2) = 383
    U12_A(x1,x2) = 383
    U111_A(x1,x2,x3) = 383
    U112_A(x1,x2,x3) = 383
    U113_A(x1,x2,x3) = 383
    U114_A(x1,x2) = 383
    s_A(x1) = 383
    length_A(x1) = 383
    U13_A(x1) = 383
    isNatList_A(x1) = 383
    U121_A(x1,x2) = 383
    U122_A(x1) = 383
    U131_A(x1,x2,x3,x4) = 383
    U132_A(x1,x2,x3,x4) = 383
    U133_A(x1,x2,x3,x4) = 383
    U134_A(x1,x2,x3,x4) = 383
    U135_A(x1,x2,x3,x4) = 383
    U136_A(x1,x2,x3,x4) = 383
    take_A(x1,x2) = 383
    U21_A(x1,x2) = 383
    U22_A(x1,x2) = 383
    U23_A(x1) = 359
    U31_A(x1,x2) = 383
    U32_A(x1,x2) = 383
    U33_A(x1) = 383
    U41_A(x1,x2,x3) = 383
    U42_A(x1,x2,x3) = 383
    U43_A(x1,x2,x3) = 383
    U44_A(x1,x2,x3) = 383
    U45_A(x1,x2) = 383
    U46_A(x1) = 383
    U51_A(x1,x2) = 383
    U52_A(x1) = 383
    U61_A(x1,x2) = 383
    U62_A(x1) = 383
    U91_A(x1,x2,x3) = 383
    U92_A(x1,x2,x3) = 383
    U93_A(x1,x2,x3) = 383
    U94_A(x1,x2,x3) = 383
    U95_A(x1,x2) = 383
    U96_A(x1) = 383
    U71_A(x1) = 383
    U81_A(x1) = 382
    active_A(x1) = max{253, x1 - 1}
    nil_A = 255

The next rules are strictly ordered:

  p124

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: active#(zeros()) -> mark#(cons(|0|(),zeros()))
p2: mark#(cons(X1,X2)) -> mark#(X1)
p3: mark#(zeros()) -> active#(zeros())
p4: mark#(U101(X1,X2,X3)) -> active#(U101(mark(X1),X2,X3))
p5: active#(U101(tt(),V1,V2)) -> mark#(U102(isNatKind(V1),V1,V2))
p6: mark#(U101(X1,X2,X3)) -> mark#(X1)
p7: mark#(U102(X1,X2,X3)) -> active#(U102(mark(X1),X2,X3))
p8: active#(U102(tt(),V1,V2)) -> mark#(U103(isNatIListKind(V2),V1,V2))
p9: mark#(U102(X1,X2,X3)) -> mark#(X1)
p10: mark#(isNatKind(X)) -> active#(isNatKind(X))
p11: active#(U103(tt(),V1,V2)) -> mark#(U104(isNatIListKind(V2),V1,V2))
p12: mark#(U103(X1,X2,X3)) -> active#(U103(mark(X1),X2,X3))
p13: active#(U104(tt(),V1,V2)) -> mark#(U105(isNat(V1),V2))
p14: mark#(U103(X1,X2,X3)) -> mark#(X1)
p15: mark#(isNatIListKind(X)) -> active#(isNatIListKind(X))
p16: active#(U105(tt(),V2)) -> mark#(U106(isNatIList(V2)))
p17: mark#(U104(X1,X2,X3)) -> active#(U104(mark(X1),X2,X3))
p18: active#(U11(tt(),V1)) -> mark#(U12(isNatIListKind(V1),V1))
p19: mark#(U104(X1,X2,X3)) -> mark#(X1)
p20: mark#(U105(X1,X2)) -> active#(U105(mark(X1),X2))
p21: active#(U111(tt(),L,N)) -> mark#(U112(isNatIListKind(L),L,N))
p22: mark#(U105(X1,X2)) -> mark#(X1)
p23: mark#(isNat(X)) -> active#(isNat(X))
p24: active#(U112(tt(),L,N)) -> mark#(U113(isNat(N),L,N))
p25: mark#(U106(X)) -> mark#(X)
p26: mark#(isNatIList(X)) -> active#(isNatIList(X))
p27: active#(U113(tt(),L,N)) -> mark#(U114(isNatKind(N),L))
p28: mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2))
p29: active#(U114(tt(),L)) -> mark#(s(length(L)))
p30: mark#(U11(X1,X2)) -> mark#(X1)
p31: mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2))
p32: active#(U12(tt(),V1)) -> mark#(U13(isNatList(V1)))
p33: mark#(U12(X1,X2)) -> mark#(X1)
p34: mark#(U111(X1,X2,X3)) -> active#(U111(mark(X1),X2,X3))
p35: active#(U121(tt(),IL)) -> mark#(U122(isNatIListKind(IL)))
p36: mark#(U111(X1,X2,X3)) -> mark#(X1)
p37: mark#(U112(X1,X2,X3)) -> active#(U112(mark(X1),X2,X3))
p38: active#(U131(tt(),IL,M,N)) -> mark#(U132(isNatIListKind(IL),IL,M,N))
p39: mark#(U112(X1,X2,X3)) -> mark#(X1)
p40: mark#(U113(X1,X2,X3)) -> active#(U113(mark(X1),X2,X3))
p41: active#(U132(tt(),IL,M,N)) -> mark#(U133(isNat(M),IL,M,N))
p42: mark#(U113(X1,X2,X3)) -> mark#(X1)
p43: mark#(U114(X1,X2)) -> active#(U114(mark(X1),X2))
p44: active#(U133(tt(),IL,M,N)) -> mark#(U134(isNatKind(M),IL,M,N))
p45: mark#(U114(X1,X2)) -> mark#(X1)
p46: mark#(s(X)) -> active#(s(mark(X)))
p47: active#(U134(tt(),IL,M,N)) -> mark#(U135(isNat(N),IL,M,N))
p48: mark#(s(X)) -> mark#(X)
p49: mark#(length(X)) -> active#(length(mark(X)))
p50: active#(U135(tt(),IL,M,N)) -> mark#(U136(isNatKind(N),IL,M,N))
p51: mark#(length(X)) -> mark#(X)
p52: mark#(U13(X)) -> mark#(X)
p53: mark#(isNatList(X)) -> active#(isNatList(X))
p54: active#(U136(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL)))
p55: mark#(U121(X1,X2)) -> active#(U121(mark(X1),X2))
p56: active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1))
p57: mark#(U121(X1,X2)) -> mark#(X1)
p58: mark#(U122(X)) -> mark#(X)
p59: mark#(U131(X1,X2,X3,X4)) -> active#(U131(mark(X1),X2,X3,X4))
p60: active#(U22(tt(),V1)) -> mark#(U23(isNat(V1)))
p61: mark#(U131(X1,X2,X3,X4)) -> mark#(X1)
p62: mark#(U132(X1,X2,X3,X4)) -> active#(U132(mark(X1),X2,X3,X4))
p63: active#(U31(tt(),V)) -> mark#(U32(isNatIListKind(V),V))
p64: mark#(U132(X1,X2,X3,X4)) -> mark#(X1)
p65: mark#(U133(X1,X2,X3,X4)) -> active#(U133(mark(X1),X2,X3,X4))
p66: active#(U32(tt(),V)) -> mark#(U33(isNatList(V)))
p67: mark#(U133(X1,X2,X3,X4)) -> mark#(X1)
p68: mark#(U134(X1,X2,X3,X4)) -> active#(U134(mark(X1),X2,X3,X4))
p69: active#(U41(tt(),V1,V2)) -> mark#(U42(isNatKind(V1),V1,V2))
p70: mark#(U134(X1,X2,X3,X4)) -> mark#(X1)
p71: mark#(U135(X1,X2,X3,X4)) -> active#(U135(mark(X1),X2,X3,X4))
p72: active#(U42(tt(),V1,V2)) -> mark#(U43(isNatIListKind(V2),V1,V2))
p73: mark#(U135(X1,X2,X3,X4)) -> mark#(X1)
p74: mark#(U136(X1,X2,X3,X4)) -> active#(U136(mark(X1),X2,X3,X4))
p75: active#(U43(tt(),V1,V2)) -> mark#(U44(isNatIListKind(V2),V1,V2))
p76: mark#(U136(X1,X2,X3,X4)) -> mark#(X1)
p77: mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2)))
p78: active#(U44(tt(),V1,V2)) -> mark#(U45(isNat(V1),V2))
p79: mark#(take(X1,X2)) -> mark#(X1)
p80: mark#(take(X1,X2)) -> mark#(X2)
p81: mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2))
p82: active#(U45(tt(),V2)) -> mark#(U46(isNatIList(V2)))
p83: mark#(U21(X1,X2)) -> mark#(X1)
p84: mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2))
p85: active#(U51(tt(),V2)) -> mark#(U52(isNatIListKind(V2)))
p86: mark#(U22(X1,X2)) -> mark#(X1)
p87: mark#(U23(X)) -> mark#(X)
p88: mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2))
p89: active#(U61(tt(),V2)) -> mark#(U62(isNatIListKind(V2)))
p90: mark#(U31(X1,X2)) -> mark#(X1)
p91: mark#(U32(X1,X2)) -> active#(U32(mark(X1),X2))
p92: active#(U91(tt(),V1,V2)) -> mark#(U92(isNatKind(V1),V1,V2))
p93: mark#(U32(X1,X2)) -> mark#(X1)
p94: mark#(U33(X)) -> mark#(X)
p95: mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3))
p96: active#(U92(tt(),V1,V2)) -> mark#(U93(isNatIListKind(V2),V1,V2))
p97: mark#(U41(X1,X2,X3)) -> mark#(X1)
p98: mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3))
p99: active#(U93(tt(),V1,V2)) -> mark#(U94(isNatIListKind(V2),V1,V2))
p100: mark#(U42(X1,X2,X3)) -> mark#(X1)
p101: mark#(U43(X1,X2,X3)) -> active#(U43(mark(X1),X2,X3))
p102: active#(U94(tt(),V1,V2)) -> mark#(U95(isNat(V1),V2))
p103: mark#(U43(X1,X2,X3)) -> mark#(X1)
p104: mark#(U44(X1,X2,X3)) -> active#(U44(mark(X1),X2,X3))
p105: active#(U95(tt(),V2)) -> mark#(U96(isNatList(V2)))
p106: mark#(U44(X1,X2,X3)) -> mark#(X1)
p107: mark#(U45(X1,X2)) -> active#(U45(mark(X1),X2))
p108: active#(isNat(length(V1))) -> mark#(U11(isNatIListKind(V1),V1))
p109: mark#(U45(X1,X2)) -> mark#(X1)
p110: mark#(U46(X)) -> active#(U46(mark(X)))
p111: active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1))
p112: mark#(U46(X)) -> mark#(X)
p113: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p114: active#(isNatIList(V)) -> mark#(U31(isNatIListKind(V),V))
p115: mark#(U51(X1,X2)) -> mark#(X1)
p116: mark#(U52(X)) -> mark#(X)
p117: mark#(U61(X1,X2)) -> active#(U61(mark(X1),X2))
p118: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNatKind(V1),V1,V2))
p119: mark#(U61(X1,X2)) -> mark#(X1)
p120: mark#(U62(X)) -> active#(U62(mark(X)))
p121: active#(isNatIListKind(cons(V1,V2))) -> mark#(U51(isNatKind(V1),V2))
p122: mark#(U62(X)) -> mark#(X)
p123: mark#(U71(X)) -> mark#(X)
p124: active#(isNatIListKind(take(V1,V2))) -> mark#(U61(isNatKind(V1),V2))
p125: mark#(U81(X)) -> mark#(X)
p126: mark#(U91(X1,X2,X3)) -> active#(U91(mark(X1),X2,X3))
p127: active#(isNatKind(length(V1))) -> mark#(U71(isNatIListKind(V1)))
p128: mark#(U91(X1,X2,X3)) -> mark#(X1)
p129: mark#(U92(X1,X2,X3)) -> active#(U92(mark(X1),X2,X3))
p130: active#(isNatKind(s(V1))) -> mark#(U81(isNatKind(V1)))
p131: mark#(U92(X1,X2,X3)) -> mark#(X1)
p132: mark#(U93(X1,X2,X3)) -> active#(U93(mark(X1),X2,X3))
p133: active#(isNatList(cons(V1,V2))) -> mark#(U91(isNatKind(V1),V1,V2))
p134: mark#(U93(X1,X2,X3)) -> mark#(X1)
p135: mark#(U94(X1,X2,X3)) -> active#(U94(mark(X1),X2,X3))
p136: active#(isNatList(take(V1,V2))) -> mark#(U101(isNatKind(V1),V1,V2))
p137: mark#(U94(X1,X2,X3)) -> mark#(X1)
p138: mark#(U95(X1,X2)) -> active#(U95(mark(X1),X2))
p139: active#(length(cons(N,L))) -> mark#(U111(isNatList(L),L,N))
p140: mark#(U95(X1,X2)) -> mark#(X1)
p141: mark#(U96(X)) -> mark#(X)
p142: active#(take(|0|(),IL)) -> mark#(U121(isNatIList(IL),IL))
p143: active#(take(s(M),cons(N,IL))) -> mark#(U131(isNatIList(IL),IL,M,N))

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19, p20, p21, p22, p23, p24, p25, p26, p27, p28, p29, p30, p31, p32, p33, p34, p35, p36, p37, p38, p39, p40, p41, p42, p43, p44, p45, p46, p47, p48, p49, p50, p51, p52, p53, p54, p55, p56, p57, p58, p59, p60, p61, p62, p63, p64, p65, p66, p67, p68, p69, p70, p71, p72, p73, p74, p75, p76, p77, p78, p79, p80, p81, p82, p83, p84, p85, p86, p87, p88, p89, p90, p91, p92, p93, p94, p95, p96, p97, p98, p99, p100, p101, p102, p103, p104, p105, p106, p107, p108, p109, p110, p111, p112, p113, p114, p115, p116, p117, p118, p119, p120, p121, p122, p123, p124, p125, p126, p127, p128, p129, p130, p131, p132, p133, p134, p135, p136, p137, p138, p139, p140, p141, p142, p143}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: active#(zeros()) -> mark#(cons(|0|(),zeros()))
p2: mark#(cons(X1,X2)) -> mark#(X1)
p3: mark#(U96(X)) -> mark#(X)
p4: mark#(U95(X1,X2)) -> mark#(X1)
p5: mark#(U95(X1,X2)) -> active#(U95(mark(X1),X2))
p6: active#(take(s(M),cons(N,IL))) -> mark#(U131(isNatIList(IL),IL,M,N))
p7: mark#(U94(X1,X2,X3)) -> mark#(X1)
p8: mark#(U94(X1,X2,X3)) -> active#(U94(mark(X1),X2,X3))
p9: active#(take(|0|(),IL)) -> mark#(U121(isNatIList(IL),IL))
p10: mark#(U93(X1,X2,X3)) -> mark#(X1)
p11: mark#(U93(X1,X2,X3)) -> active#(U93(mark(X1),X2,X3))
p12: active#(length(cons(N,L))) -> mark#(U111(isNatList(L),L,N))
p13: mark#(U92(X1,X2,X3)) -> mark#(X1)
p14: mark#(U92(X1,X2,X3)) -> active#(U92(mark(X1),X2,X3))
p15: active#(isNatList(take(V1,V2))) -> mark#(U101(isNatKind(V1),V1,V2))
p16: mark#(U91(X1,X2,X3)) -> mark#(X1)
p17: mark#(U91(X1,X2,X3)) -> active#(U91(mark(X1),X2,X3))
p18: active#(isNatList(cons(V1,V2))) -> mark#(U91(isNatKind(V1),V1,V2))
p19: mark#(U81(X)) -> mark#(X)
p20: mark#(U71(X)) -> mark#(X)
p21: mark#(U62(X)) -> mark#(X)
p22: mark#(U62(X)) -> active#(U62(mark(X)))
p23: active#(isNatKind(s(V1))) -> mark#(U81(isNatKind(V1)))
p24: mark#(U61(X1,X2)) -> mark#(X1)
p25: mark#(U61(X1,X2)) -> active#(U61(mark(X1),X2))
p26: active#(isNatKind(length(V1))) -> mark#(U71(isNatIListKind(V1)))
p27: mark#(U52(X)) -> mark#(X)
p28: mark#(U51(X1,X2)) -> mark#(X1)
p29: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p30: active#(isNatIListKind(take(V1,V2))) -> mark#(U61(isNatKind(V1),V2))
p31: mark#(U46(X)) -> mark#(X)
p32: mark#(U46(X)) -> active#(U46(mark(X)))
p33: active#(isNatIListKind(cons(V1,V2))) -> mark#(U51(isNatKind(V1),V2))
p34: mark#(U45(X1,X2)) -> mark#(X1)
p35: mark#(U45(X1,X2)) -> active#(U45(mark(X1),X2))
p36: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNatKind(V1),V1,V2))
p37: mark#(U44(X1,X2,X3)) -> mark#(X1)
p38: mark#(U44(X1,X2,X3)) -> active#(U44(mark(X1),X2,X3))
p39: active#(isNatIList(V)) -> mark#(U31(isNatIListKind(V),V))
p40: mark#(U43(X1,X2,X3)) -> mark#(X1)
p41: mark#(U43(X1,X2,X3)) -> active#(U43(mark(X1),X2,X3))
p42: active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1))
p43: mark#(U42(X1,X2,X3)) -> mark#(X1)
p44: mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3))
p45: active#(isNat(length(V1))) -> mark#(U11(isNatIListKind(V1),V1))
p46: mark#(U41(X1,X2,X3)) -> mark#(X1)
p47: mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3))
p48: active#(U95(tt(),V2)) -> mark#(U96(isNatList(V2)))
p49: mark#(U33(X)) -> mark#(X)
p50: mark#(U32(X1,X2)) -> mark#(X1)
p51: mark#(U32(X1,X2)) -> active#(U32(mark(X1),X2))
p52: active#(U94(tt(),V1,V2)) -> mark#(U95(isNat(V1),V2))
p53: mark#(U31(X1,X2)) -> mark#(X1)
p54: mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2))
p55: active#(U93(tt(),V1,V2)) -> mark#(U94(isNatIListKind(V2),V1,V2))
p56: mark#(U23(X)) -> mark#(X)
p57: mark#(U22(X1,X2)) -> mark#(X1)
p58: mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2))
p59: active#(U92(tt(),V1,V2)) -> mark#(U93(isNatIListKind(V2),V1,V2))
p60: mark#(U21(X1,X2)) -> mark#(X1)
p61: mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2))
p62: active#(U91(tt(),V1,V2)) -> mark#(U92(isNatKind(V1),V1,V2))
p63: mark#(take(X1,X2)) -> mark#(X2)
p64: mark#(take(X1,X2)) -> mark#(X1)
p65: mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2)))
p66: active#(U61(tt(),V2)) -> mark#(U62(isNatIListKind(V2)))
p67: mark#(U136(X1,X2,X3,X4)) -> mark#(X1)
p68: mark#(U136(X1,X2,X3,X4)) -> active#(U136(mark(X1),X2,X3,X4))
p69: active#(U51(tt(),V2)) -> mark#(U52(isNatIListKind(V2)))
p70: mark#(U135(X1,X2,X3,X4)) -> mark#(X1)
p71: mark#(U135(X1,X2,X3,X4)) -> active#(U135(mark(X1),X2,X3,X4))
p72: active#(U45(tt(),V2)) -> mark#(U46(isNatIList(V2)))
p73: mark#(U134(X1,X2,X3,X4)) -> mark#(X1)
p74: mark#(U134(X1,X2,X3,X4)) -> active#(U134(mark(X1),X2,X3,X4))
p75: active#(U44(tt(),V1,V2)) -> mark#(U45(isNat(V1),V2))
p76: mark#(U133(X1,X2,X3,X4)) -> mark#(X1)
p77: mark#(U133(X1,X2,X3,X4)) -> active#(U133(mark(X1),X2,X3,X4))
p78: active#(U43(tt(),V1,V2)) -> mark#(U44(isNatIListKind(V2),V1,V2))
p79: mark#(U132(X1,X2,X3,X4)) -> mark#(X1)
p80: mark#(U132(X1,X2,X3,X4)) -> active#(U132(mark(X1),X2,X3,X4))
p81: active#(U42(tt(),V1,V2)) -> mark#(U43(isNatIListKind(V2),V1,V2))
p82: mark#(U131(X1,X2,X3,X4)) -> mark#(X1)
p83: mark#(U131(X1,X2,X3,X4)) -> active#(U131(mark(X1),X2,X3,X4))
p84: active#(U41(tt(),V1,V2)) -> mark#(U42(isNatKind(V1),V1,V2))
p85: mark#(U122(X)) -> mark#(X)
p86: mark#(U121(X1,X2)) -> mark#(X1)
p87: mark#(U121(X1,X2)) -> active#(U121(mark(X1),X2))
p88: active#(U32(tt(),V)) -> mark#(U33(isNatList(V)))
p89: mark#(isNatList(X)) -> active#(isNatList(X))
p90: active#(U31(tt(),V)) -> mark#(U32(isNatIListKind(V),V))
p91: mark#(U13(X)) -> mark#(X)
p92: mark#(length(X)) -> mark#(X)
p93: mark#(length(X)) -> active#(length(mark(X)))
p94: active#(U22(tt(),V1)) -> mark#(U23(isNat(V1)))
p95: mark#(s(X)) -> mark#(X)
p96: mark#(s(X)) -> active#(s(mark(X)))
p97: active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1))
p98: mark#(U114(X1,X2)) -> mark#(X1)
p99: mark#(U114(X1,X2)) -> active#(U114(mark(X1),X2))
p100: active#(U136(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL)))
p101: mark#(U113(X1,X2,X3)) -> mark#(X1)
p102: mark#(U113(X1,X2,X3)) -> active#(U113(mark(X1),X2,X3))
p103: active#(U135(tt(),IL,M,N)) -> mark#(U136(isNatKind(N),IL,M,N))
p104: mark#(U112(X1,X2,X3)) -> mark#(X1)
p105: mark#(U112(X1,X2,X3)) -> active#(U112(mark(X1),X2,X3))
p106: active#(U134(tt(),IL,M,N)) -> mark#(U135(isNat(N),IL,M,N))
p107: mark#(U111(X1,X2,X3)) -> mark#(X1)
p108: mark#(U111(X1,X2,X3)) -> active#(U111(mark(X1),X2,X3))
p109: active#(U133(tt(),IL,M,N)) -> mark#(U134(isNatKind(M),IL,M,N))
p110: mark#(U12(X1,X2)) -> mark#(X1)
p111: mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2))
p112: active#(U132(tt(),IL,M,N)) -> mark#(U133(isNat(M),IL,M,N))
p113: mark#(U11(X1,X2)) -> mark#(X1)
p114: mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2))
p115: active#(U131(tt(),IL,M,N)) -> mark#(U132(isNatIListKind(IL),IL,M,N))
p116: mark#(isNatIList(X)) -> active#(isNatIList(X))
p117: active#(U121(tt(),IL)) -> mark#(U122(isNatIListKind(IL)))
p118: mark#(U106(X)) -> mark#(X)
p119: mark#(isNat(X)) -> active#(isNat(X))
p120: active#(U12(tt(),V1)) -> mark#(U13(isNatList(V1)))
p121: mark#(U105(X1,X2)) -> mark#(X1)
p122: mark#(U105(X1,X2)) -> active#(U105(mark(X1),X2))
p123: active#(U114(tt(),L)) -> mark#(s(length(L)))
p124: mark#(U104(X1,X2,X3)) -> mark#(X1)
p125: mark#(U104(X1,X2,X3)) -> active#(U104(mark(X1),X2,X3))
p126: active#(U113(tt(),L,N)) -> mark#(U114(isNatKind(N),L))
p127: mark#(isNatIListKind(X)) -> active#(isNatIListKind(X))
p128: active#(U112(tt(),L,N)) -> mark#(U113(isNat(N),L,N))
p129: mark#(U103(X1,X2,X3)) -> mark#(X1)
p130: mark#(U103(X1,X2,X3)) -> active#(U103(mark(X1),X2,X3))
p131: active#(U111(tt(),L,N)) -> mark#(U112(isNatIListKind(L),L,N))
p132: mark#(isNatKind(X)) -> active#(isNatKind(X))
p133: active#(U11(tt(),V1)) -> mark#(U12(isNatIListKind(V1),V1))
p134: mark#(U102(X1,X2,X3)) -> mark#(X1)
p135: mark#(U102(X1,X2,X3)) -> active#(U102(mark(X1),X2,X3))
p136: active#(U105(tt(),V2)) -> mark#(U106(isNatIList(V2)))
p137: mark#(U101(X1,X2,X3)) -> mark#(X1)
p138: mark#(U101(X1,X2,X3)) -> active#(U101(mark(X1),X2,X3))
p139: active#(U104(tt(),V1,V2)) -> mark#(U105(isNat(V1),V2))
p140: mark#(zeros()) -> active#(zeros())
p141: active#(U103(tt(),V1,V2)) -> mark#(U104(isNatIListKind(V2),V1,V2))
p142: active#(U102(tt(),V1,V2)) -> mark#(U103(isNatIListKind(V2),V1,V2))
p143: active#(U101(tt(),V1,V2)) -> mark#(U102(isNatKind(V1),V1,V2))

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41, r42, r43, r44, r45, r46, r47, r48, r49, r50, r51, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76, r77, r78, r79, r80, r81, r82, r83, r84, r85, r86, r87, r88, r89, r90, r91, r92, r93, r94, r95, r96, r97, r98, r99, r100, r101, r102, r103, r104, r105, r106, r107, r108, r109, r110, r111, r112, r113, r114, r115, r116, r117, r118, r119, r120, r121, r122, r123, r124, r125, r126, r127, r128, r129, r130, r131, r132, r133, r134, r135, r136, r137, r138, r139, r140, r141, r142, r143, r144, r145, r146, r147, r148, r149, r150, r151, r152, r153, r154, r155, r156, r157, r158, r159, r160, r161, r162, r163, r164, r165, r166, r167, r168, r169, r170, r171, r172, r173, r174, r175, r176, r177, r178, r179, r180, r181, r182, r183, r184, r185, r186, r187, r188, r189, r190, r191, r192, r193, r194, r195, r196, r197, r198, r199, r200, r201, r202, r203, r204, r205, r206, r207, r208, r209, r210, r211, r212, r213, r214, r215, r216, r217, r218, r219, r220, r221, r222, r223, r224, r225, r226, r227, r228, r229, r230, r231, r232, r233, r234, r235, r236, r237, r238, r239, r240, r241, r242, r243, r244, r245, r246, r247, r248, r249, r250, r251, r252, r253, r254, r255, r256, r257, r258, r259, r260, r261, r262, r263, r264, r265, r266, r267, r268, r269, r270, r271, r272, r273, r274, r275, r276, r277, r278, r279, r280, r281, r282, r283, r284, r285, r286, r287, r288, r289, r290, r291, r292, r293, r294, r295, r296, r297, r298, r299, r300, r301, r302, r303, r304, r305, r306, r307, r308, r309, r310, r311, r312, r313, r314, r315, r316, r317, r318, r319, r320, r321, r322, r323, r324, r325, r326, r327, r328, r329, r330, r331, r332, r333, r334, r335, r336, r337, r338, r339, r340, r341, r342, r343, r344, r345, r346, r347, r348, r349, r350, r351, r352, r353, r354, r355, r356, r357, r358

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    active#_A(x1) = max{0, x1 - 22}
    zeros_A = 96
    mark#_A(x1) = max{0, x1 - 22}
    cons_A(x1,x2) = max{96, x1 + 79, x2}
    |0|_A = 2
    U96_A(x1) = max{11, x1}
    U95_A(x1,x2) = max{12, x1}
    mark_A(x1) = max{17, x1}
    take_A(x1,x2) = max{98, x1 + 81, x2 + 63}
    s_A(x1) = max{29, x1}
    U131_A(x1,x2,x3,x4) = max{159, x1, x2 + 63, x3 + 81, x4 + 141}
    isNatIList_A(x1) = 1
    U94_A(x1,x2,x3) = max{13, x1}
    U121_A(x1,x2) = max{82, x1 + 65, x2 + 63}
    U93_A(x1,x2,x3) = max{12, x1}
    length_A(x1) = max{78, x1}
    U111_A(x1,x2,x3) = max{x1 + 61, x2, x3 + 78}
    isNatList_A(x1) = 0
    U92_A(x1,x2,x3) = max{11, x1}
    U101_A(x1,x2,x3) = max{7, x1}
    isNatKind_A(x1) = 3
    U91_A(x1,x2,x3) = max{1, x1}
    U81_A(x1) = max{3, x1}
    U71_A(x1) = max{8, x1}
    U62_A(x1) = max{1, x1}
    U61_A(x1,x2) = max{4, x1}
    isNatIListKind_A(x1) = 4
    U52_A(x1) = max{4, x1}
    U51_A(x1,x2) = max{5, x1}
    U46_A(x1) = max{10, x1}
    U45_A(x1,x2) = max{9, x1}
    U41_A(x1,x2,x3) = max{2, x1}
    U44_A(x1,x2,x3) = max{8, x1}
    U31_A(x1,x2) = max{5, x1}
    U43_A(x1,x2,x3) = max{11, x1}
    isNat_A(x1) = 14
    U21_A(x1,x2) = max{17, x1}
    U42_A(x1,x2,x3) = max{9, x1}
    U11_A(x1,x2) = max{5, x1}
    tt_A = 10
    U33_A(x1) = max{17, x1}
    U32_A(x1,x2) = max{11, x1}
    U23_A(x1) = max{10, x1}
    U22_A(x1,x2) = max{10, x1}
    U136_A(x1,x2,x3,x4) = max{x1 + 89, x2 + 63, x3 + 81, x4 + 107}
    U135_A(x1,x2,x3,x4) = max{x1 + 98, x2 + 63, x3 + 81, x4 + 136}
    U134_A(x1,x2,x3,x4) = max{x1 + 118, x2 + 63, x3 + 81, x4 + 136}
    U133_A(x1,x2,x3,x4) = max{x1 + 112, x2 + 63, x3 + 81, x4 + 139}
    U132_A(x1,x2,x3,x4) = max{140, x1 + 121, x2 + 63, x3 + 81, x4 + 139}
    U122_A(x1) = max{82, x1}
    U13_A(x1) = max{17, x1}
    U114_A(x1,x2) = max{78, x1 + 20, x2}
    U113_A(x1,x2,x3) = max{x1 + 60, x2, x3 + 78}
    U112_A(x1,x2,x3) = max{x1 + 17, x2, x3 + 78}
    U12_A(x1,x2) = max{1, x1}
    U106_A(x1) = max{11, x1}
    U105_A(x1,x2) = max{16, x1}
    U104_A(x1,x2,x3) = max{15, x1}
    U103_A(x1,x2,x3) = max{9, x1}
    U102_A(x1,x2,x3) = max{8, x1}
    active_A(x1) = max{17, x1}
    nil_A = 18

The next rules are strictly ordered:

  p2, p63, p64, p67, p70, p73, p76, p79, p86, p98, p101, p104, p107

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: active#(zeros()) -> mark#(cons(|0|(),zeros()))
p2: mark#(U96(X)) -> mark#(X)
p3: mark#(U95(X1,X2)) -> mark#(X1)
p4: mark#(U95(X1,X2)) -> active#(U95(mark(X1),X2))
p5: active#(take(s(M),cons(N,IL))) -> mark#(U131(isNatIList(IL),IL,M,N))
p6: mark#(U94(X1,X2,X3)) -> mark#(X1)
p7: mark#(U94(X1,X2,X3)) -> active#(U94(mark(X1),X2,X3))
p8: active#(take(|0|(),IL)) -> mark#(U121(isNatIList(IL),IL))
p9: mark#(U93(X1,X2,X3)) -> mark#(X1)
p10: mark#(U93(X1,X2,X3)) -> active#(U93(mark(X1),X2,X3))
p11: active#(length(cons(N,L))) -> mark#(U111(isNatList(L),L,N))
p12: mark#(U92(X1,X2,X3)) -> mark#(X1)
p13: mark#(U92(X1,X2,X3)) -> active#(U92(mark(X1),X2,X3))
p14: active#(isNatList(take(V1,V2))) -> mark#(U101(isNatKind(V1),V1,V2))
p15: mark#(U91(X1,X2,X3)) -> mark#(X1)
p16: mark#(U91(X1,X2,X3)) -> active#(U91(mark(X1),X2,X3))
p17: active#(isNatList(cons(V1,V2))) -> mark#(U91(isNatKind(V1),V1,V2))
p18: mark#(U81(X)) -> mark#(X)
p19: mark#(U71(X)) -> mark#(X)
p20: mark#(U62(X)) -> mark#(X)
p21: mark#(U62(X)) -> active#(U62(mark(X)))
p22: active#(isNatKind(s(V1))) -> mark#(U81(isNatKind(V1)))
p23: mark#(U61(X1,X2)) -> mark#(X1)
p24: mark#(U61(X1,X2)) -> active#(U61(mark(X1),X2))
p25: active#(isNatKind(length(V1))) -> mark#(U71(isNatIListKind(V1)))
p26: mark#(U52(X)) -> mark#(X)
p27: mark#(U51(X1,X2)) -> mark#(X1)
p28: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p29: active#(isNatIListKind(take(V1,V2))) -> mark#(U61(isNatKind(V1),V2))
p30: mark#(U46(X)) -> mark#(X)
p31: mark#(U46(X)) -> active#(U46(mark(X)))
p32: active#(isNatIListKind(cons(V1,V2))) -> mark#(U51(isNatKind(V1),V2))
p33: mark#(U45(X1,X2)) -> mark#(X1)
p34: mark#(U45(X1,X2)) -> active#(U45(mark(X1),X2))
p35: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNatKind(V1),V1,V2))
p36: mark#(U44(X1,X2,X3)) -> mark#(X1)
p37: mark#(U44(X1,X2,X3)) -> active#(U44(mark(X1),X2,X3))
p38: active#(isNatIList(V)) -> mark#(U31(isNatIListKind(V),V))
p39: mark#(U43(X1,X2,X3)) -> mark#(X1)
p40: mark#(U43(X1,X2,X3)) -> active#(U43(mark(X1),X2,X3))
p41: active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1))
p42: mark#(U42(X1,X2,X3)) -> mark#(X1)
p43: mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3))
p44: active#(isNat(length(V1))) -> mark#(U11(isNatIListKind(V1),V1))
p45: mark#(U41(X1,X2,X3)) -> mark#(X1)
p46: mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3))
p47: active#(U95(tt(),V2)) -> mark#(U96(isNatList(V2)))
p48: mark#(U33(X)) -> mark#(X)
p49: mark#(U32(X1,X2)) -> mark#(X1)
p50: mark#(U32(X1,X2)) -> active#(U32(mark(X1),X2))
p51: active#(U94(tt(),V1,V2)) -> mark#(U95(isNat(V1),V2))
p52: mark#(U31(X1,X2)) -> mark#(X1)
p53: mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2))
p54: active#(U93(tt(),V1,V2)) -> mark#(U94(isNatIListKind(V2),V1,V2))
p55: mark#(U23(X)) -> mark#(X)
p56: mark#(U22(X1,X2)) -> mark#(X1)
p57: mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2))
p58: active#(U92(tt(),V1,V2)) -> mark#(U93(isNatIListKind(V2),V1,V2))
p59: mark#(U21(X1,X2)) -> mark#(X1)
p60: mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2))
p61: active#(U91(tt(),V1,V2)) -> mark#(U92(isNatKind(V1),V1,V2))
p62: mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2)))
p63: active#(U61(tt(),V2)) -> mark#(U62(isNatIListKind(V2)))
p64: mark#(U136(X1,X2,X3,X4)) -> active#(U136(mark(X1),X2,X3,X4))
p65: active#(U51(tt(),V2)) -> mark#(U52(isNatIListKind(V2)))
p66: mark#(U135(X1,X2,X3,X4)) -> active#(U135(mark(X1),X2,X3,X4))
p67: active#(U45(tt(),V2)) -> mark#(U46(isNatIList(V2)))
p68: mark#(U134(X1,X2,X3,X4)) -> active#(U134(mark(X1),X2,X3,X4))
p69: active#(U44(tt(),V1,V2)) -> mark#(U45(isNat(V1),V2))
p70: mark#(U133(X1,X2,X3,X4)) -> active#(U133(mark(X1),X2,X3,X4))
p71: active#(U43(tt(),V1,V2)) -> mark#(U44(isNatIListKind(V2),V1,V2))
p72: mark#(U132(X1,X2,X3,X4)) -> active#(U132(mark(X1),X2,X3,X4))
p73: active#(U42(tt(),V1,V2)) -> mark#(U43(isNatIListKind(V2),V1,V2))
p74: mark#(U131(X1,X2,X3,X4)) -> mark#(X1)
p75: mark#(U131(X1,X2,X3,X4)) -> active#(U131(mark(X1),X2,X3,X4))
p76: active#(U41(tt(),V1,V2)) -> mark#(U42(isNatKind(V1),V1,V2))
p77: mark#(U122(X)) -> mark#(X)
p78: mark#(U121(X1,X2)) -> active#(U121(mark(X1),X2))
p79: active#(U32(tt(),V)) -> mark#(U33(isNatList(V)))
p80: mark#(isNatList(X)) -> active#(isNatList(X))
p81: active#(U31(tt(),V)) -> mark#(U32(isNatIListKind(V),V))
p82: mark#(U13(X)) -> mark#(X)
p83: mark#(length(X)) -> mark#(X)
p84: mark#(length(X)) -> active#(length(mark(X)))
p85: active#(U22(tt(),V1)) -> mark#(U23(isNat(V1)))
p86: mark#(s(X)) -> mark#(X)
p87: mark#(s(X)) -> active#(s(mark(X)))
p88: active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1))
p89: mark#(U114(X1,X2)) -> active#(U114(mark(X1),X2))
p90: active#(U136(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL)))
p91: mark#(U113(X1,X2,X3)) -> active#(U113(mark(X1),X2,X3))
p92: active#(U135(tt(),IL,M,N)) -> mark#(U136(isNatKind(N),IL,M,N))
p93: mark#(U112(X1,X2,X3)) -> active#(U112(mark(X1),X2,X3))
p94: active#(U134(tt(),IL,M,N)) -> mark#(U135(isNat(N),IL,M,N))
p95: mark#(U111(X1,X2,X3)) -> active#(U111(mark(X1),X2,X3))
p96: active#(U133(tt(),IL,M,N)) -> mark#(U134(isNatKind(M),IL,M,N))
p97: mark#(U12(X1,X2)) -> mark#(X1)
p98: mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2))
p99: active#(U132(tt(),IL,M,N)) -> mark#(U133(isNat(M),IL,M,N))
p100: mark#(U11(X1,X2)) -> mark#(X1)
p101: mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2))
p102: active#(U131(tt(),IL,M,N)) -> mark#(U132(isNatIListKind(IL),IL,M,N))
p103: mark#(isNatIList(X)) -> active#(isNatIList(X))
p104: active#(U121(tt(),IL)) -> mark#(U122(isNatIListKind(IL)))
p105: mark#(U106(X)) -> mark#(X)
p106: mark#(isNat(X)) -> active#(isNat(X))
p107: active#(U12(tt(),V1)) -> mark#(U13(isNatList(V1)))
p108: mark#(U105(X1,X2)) -> mark#(X1)
p109: mark#(U105(X1,X2)) -> active#(U105(mark(X1),X2))
p110: active#(U114(tt(),L)) -> mark#(s(length(L)))
p111: mark#(U104(X1,X2,X3)) -> mark#(X1)
p112: mark#(U104(X1,X2,X3)) -> active#(U104(mark(X1),X2,X3))
p113: active#(U113(tt(),L,N)) -> mark#(U114(isNatKind(N),L))
p114: mark#(isNatIListKind(X)) -> active#(isNatIListKind(X))
p115: active#(U112(tt(),L,N)) -> mark#(U113(isNat(N),L,N))
p116: mark#(U103(X1,X2,X3)) -> mark#(X1)
p117: mark#(U103(X1,X2,X3)) -> active#(U103(mark(X1),X2,X3))
p118: active#(U111(tt(),L,N)) -> mark#(U112(isNatIListKind(L),L,N))
p119: mark#(isNatKind(X)) -> active#(isNatKind(X))
p120: active#(U11(tt(),V1)) -> mark#(U12(isNatIListKind(V1),V1))
p121: mark#(U102(X1,X2,X3)) -> mark#(X1)
p122: mark#(U102(X1,X2,X3)) -> active#(U102(mark(X1),X2,X3))
p123: active#(U105(tt(),V2)) -> mark#(U106(isNatIList(V2)))
p124: mark#(U101(X1,X2,X3)) -> mark#(X1)
p125: mark#(U101(X1,X2,X3)) -> active#(U101(mark(X1),X2,X3))
p126: active#(U104(tt(),V1,V2)) -> mark#(U105(isNat(V1),V2))
p127: mark#(zeros()) -> active#(zeros())
p128: active#(U103(tt(),V1,V2)) -> mark#(U104(isNatIListKind(V2),V1,V2))
p129: active#(U102(tt(),V1,V2)) -> mark#(U103(isNatIListKind(V2),V1,V2))
p130: active#(U101(tt(),V1,V2)) -> mark#(U102(isNatKind(V1),V1,V2))

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19, p20, p21, p22, p23, p24, p25, p26, p27, p28, p29, p30, p31, p32, p33, p34, p35, p36, p37, p38, p39, p40, p41, p42, p43, p44, p45, p46, p47, p48, p49, p50, p51, p52, p53, p54, p55, p56, p57, p58, p59, p60, p61, p62, p63, p64, p65, p66, p67, p68, p69, p70, p71, p72, p73, p74, p75, p76, p77, p78, p79, p80, p81, p82, p83, p84, p85, p86, p87, p88, p89, p90, p91, p92, p93, p94, p95, p96, p97, p98, p99, p100, p101, p102, p103, p104, p105, p106, p107, p108, p109, p110, p111, p112, p113, p114, p115, p116, p117, p118, p119, p120, p121, p122, p123, p124, p125, p126, p128, p129, p130}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(U96(X)) -> mark#(X)
p2: mark#(U101(X1,X2,X3)) -> active#(U101(mark(X1),X2,X3))
p3: active#(U101(tt(),V1,V2)) -> mark#(U102(isNatKind(V1),V1,V2))
p4: mark#(U101(X1,X2,X3)) -> mark#(X1)
p5: mark#(U102(X1,X2,X3)) -> active#(U102(mark(X1),X2,X3))
p6: active#(U102(tt(),V1,V2)) -> mark#(U103(isNatIListKind(V2),V1,V2))
p7: mark#(U102(X1,X2,X3)) -> mark#(X1)
p8: mark#(isNatKind(X)) -> active#(isNatKind(X))
p9: active#(U103(tt(),V1,V2)) -> mark#(U104(isNatIListKind(V2),V1,V2))
p10: mark#(U103(X1,X2,X3)) -> active#(U103(mark(X1),X2,X3))
p11: active#(U104(tt(),V1,V2)) -> mark#(U105(isNat(V1),V2))
p12: mark#(U103(X1,X2,X3)) -> mark#(X1)
p13: mark#(isNatIListKind(X)) -> active#(isNatIListKind(X))
p14: active#(U105(tt(),V2)) -> mark#(U106(isNatIList(V2)))
p15: mark#(U104(X1,X2,X3)) -> active#(U104(mark(X1),X2,X3))
p16: active#(U11(tt(),V1)) -> mark#(U12(isNatIListKind(V1),V1))
p17: mark#(U104(X1,X2,X3)) -> mark#(X1)
p18: mark#(U105(X1,X2)) -> active#(U105(mark(X1),X2))
p19: active#(U111(tt(),L,N)) -> mark#(U112(isNatIListKind(L),L,N))
p20: mark#(U105(X1,X2)) -> mark#(X1)
p21: mark#(isNat(X)) -> active#(isNat(X))
p22: active#(U112(tt(),L,N)) -> mark#(U113(isNat(N),L,N))
p23: mark#(U106(X)) -> mark#(X)
p24: mark#(isNatIList(X)) -> active#(isNatIList(X))
p25: active#(U113(tt(),L,N)) -> mark#(U114(isNatKind(N),L))
p26: mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2))
p27: active#(U114(tt(),L)) -> mark#(s(length(L)))
p28: mark#(U11(X1,X2)) -> mark#(X1)
p29: mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2))
p30: active#(U12(tt(),V1)) -> mark#(U13(isNatList(V1)))
p31: mark#(U12(X1,X2)) -> mark#(X1)
p32: mark#(U111(X1,X2,X3)) -> active#(U111(mark(X1),X2,X3))
p33: active#(U121(tt(),IL)) -> mark#(U122(isNatIListKind(IL)))
p34: mark#(U112(X1,X2,X3)) -> active#(U112(mark(X1),X2,X3))
p35: active#(U131(tt(),IL,M,N)) -> mark#(U132(isNatIListKind(IL),IL,M,N))
p36: mark#(U113(X1,X2,X3)) -> active#(U113(mark(X1),X2,X3))
p37: active#(U132(tt(),IL,M,N)) -> mark#(U133(isNat(M),IL,M,N))
p38: mark#(U114(X1,X2)) -> active#(U114(mark(X1),X2))
p39: active#(U133(tt(),IL,M,N)) -> mark#(U134(isNatKind(M),IL,M,N))
p40: mark#(s(X)) -> active#(s(mark(X)))
p41: active#(U134(tt(),IL,M,N)) -> mark#(U135(isNat(N),IL,M,N))
p42: mark#(s(X)) -> mark#(X)
p43: mark#(length(X)) -> active#(length(mark(X)))
p44: active#(U135(tt(),IL,M,N)) -> mark#(U136(isNatKind(N),IL,M,N))
p45: mark#(length(X)) -> mark#(X)
p46: mark#(U13(X)) -> mark#(X)
p47: mark#(isNatList(X)) -> active#(isNatList(X))
p48: active#(U136(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL)))
p49: mark#(U121(X1,X2)) -> active#(U121(mark(X1),X2))
p50: active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1))
p51: mark#(U122(X)) -> mark#(X)
p52: mark#(U131(X1,X2,X3,X4)) -> active#(U131(mark(X1),X2,X3,X4))
p53: active#(U22(tt(),V1)) -> mark#(U23(isNat(V1)))
p54: mark#(U131(X1,X2,X3,X4)) -> mark#(X1)
p55: mark#(U132(X1,X2,X3,X4)) -> active#(U132(mark(X1),X2,X3,X4))
p56: active#(U31(tt(),V)) -> mark#(U32(isNatIListKind(V),V))
p57: mark#(U133(X1,X2,X3,X4)) -> active#(U133(mark(X1),X2,X3,X4))
p58: active#(U32(tt(),V)) -> mark#(U33(isNatList(V)))
p59: mark#(U134(X1,X2,X3,X4)) -> active#(U134(mark(X1),X2,X3,X4))
p60: active#(U41(tt(),V1,V2)) -> mark#(U42(isNatKind(V1),V1,V2))
p61: mark#(U135(X1,X2,X3,X4)) -> active#(U135(mark(X1),X2,X3,X4))
p62: active#(U42(tt(),V1,V2)) -> mark#(U43(isNatIListKind(V2),V1,V2))
p63: mark#(U136(X1,X2,X3,X4)) -> active#(U136(mark(X1),X2,X3,X4))
p64: active#(U43(tt(),V1,V2)) -> mark#(U44(isNatIListKind(V2),V1,V2))
p65: mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2)))
p66: active#(U44(tt(),V1,V2)) -> mark#(U45(isNat(V1),V2))
p67: mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2))
p68: active#(U45(tt(),V2)) -> mark#(U46(isNatIList(V2)))
p69: mark#(U21(X1,X2)) -> mark#(X1)
p70: mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2))
p71: active#(U51(tt(),V2)) -> mark#(U52(isNatIListKind(V2)))
p72: mark#(U22(X1,X2)) -> mark#(X1)
p73: mark#(U23(X)) -> mark#(X)
p74: mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2))
p75: active#(U61(tt(),V2)) -> mark#(U62(isNatIListKind(V2)))
p76: mark#(U31(X1,X2)) -> mark#(X1)
p77: mark#(U32(X1,X2)) -> active#(U32(mark(X1),X2))
p78: active#(U91(tt(),V1,V2)) -> mark#(U92(isNatKind(V1),V1,V2))
p79: mark#(U32(X1,X2)) -> mark#(X1)
p80: mark#(U33(X)) -> mark#(X)
p81: mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3))
p82: active#(U92(tt(),V1,V2)) -> mark#(U93(isNatIListKind(V2),V1,V2))
p83: mark#(U41(X1,X2,X3)) -> mark#(X1)
p84: mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3))
p85: active#(U93(tt(),V1,V2)) -> mark#(U94(isNatIListKind(V2),V1,V2))
p86: mark#(U42(X1,X2,X3)) -> mark#(X1)
p87: mark#(U43(X1,X2,X3)) -> active#(U43(mark(X1),X2,X3))
p88: active#(U94(tt(),V1,V2)) -> mark#(U95(isNat(V1),V2))
p89: mark#(U43(X1,X2,X3)) -> mark#(X1)
p90: mark#(U44(X1,X2,X3)) -> active#(U44(mark(X1),X2,X3))
p91: active#(U95(tt(),V2)) -> mark#(U96(isNatList(V2)))
p92: mark#(U44(X1,X2,X3)) -> mark#(X1)
p93: mark#(U45(X1,X2)) -> active#(U45(mark(X1),X2))
p94: active#(isNat(length(V1))) -> mark#(U11(isNatIListKind(V1),V1))
p95: mark#(U45(X1,X2)) -> mark#(X1)
p96: mark#(U46(X)) -> active#(U46(mark(X)))
p97: active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1))
p98: mark#(U46(X)) -> mark#(X)
p99: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p100: active#(isNatIList(V)) -> mark#(U31(isNatIListKind(V),V))
p101: mark#(U51(X1,X2)) -> mark#(X1)
p102: mark#(U52(X)) -> mark#(X)
p103: mark#(U61(X1,X2)) -> active#(U61(mark(X1),X2))
p104: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNatKind(V1),V1,V2))
p105: mark#(U61(X1,X2)) -> mark#(X1)
p106: mark#(U62(X)) -> active#(U62(mark(X)))
p107: active#(isNatIListKind(cons(V1,V2))) -> mark#(U51(isNatKind(V1),V2))
p108: mark#(U62(X)) -> mark#(X)
p109: mark#(U71(X)) -> mark#(X)
p110: mark#(U81(X)) -> mark#(X)
p111: mark#(U91(X1,X2,X3)) -> active#(U91(mark(X1),X2,X3))
p112: active#(isNatIListKind(take(V1,V2))) -> mark#(U61(isNatKind(V1),V2))
p113: mark#(U91(X1,X2,X3)) -> mark#(X1)
p114: mark#(U92(X1,X2,X3)) -> active#(U92(mark(X1),X2,X3))
p115: active#(isNatKind(length(V1))) -> mark#(U71(isNatIListKind(V1)))
p116: mark#(U92(X1,X2,X3)) -> mark#(X1)
p117: mark#(U93(X1,X2,X3)) -> active#(U93(mark(X1),X2,X3))
p118: active#(isNatKind(s(V1))) -> mark#(U81(isNatKind(V1)))
p119: mark#(U93(X1,X2,X3)) -> mark#(X1)
p120: mark#(U94(X1,X2,X3)) -> active#(U94(mark(X1),X2,X3))
p121: active#(isNatList(cons(V1,V2))) -> mark#(U91(isNatKind(V1),V1,V2))
p122: mark#(U94(X1,X2,X3)) -> mark#(X1)
p123: mark#(U95(X1,X2)) -> active#(U95(mark(X1),X2))
p124: active#(isNatList(take(V1,V2))) -> mark#(U101(isNatKind(V1),V1,V2))
p125: mark#(U95(X1,X2)) -> mark#(X1)
p126: active#(length(cons(N,L))) -> mark#(U111(isNatList(L),L,N))
p127: active#(take(|0|(),IL)) -> mark#(U121(isNatIList(IL),IL))
p128: active#(take(s(M),cons(N,IL))) -> mark#(U131(isNatIList(IL),IL,M,N))

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41, r42, r43, r44, r45, r46, r47, r48, r49, r50, r51, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76, r77, r78, r79, r80, r81, r82, r83, r84, r85, r86, r87, r88, r89, r90, r91, r92, r93, r94, r95, r96, r97, r98, r99, r100, r101, r102, r103, r104, r105, r106, r107, r108, r109, r110, r111, r112, r113, r114, r115, r116, r117, r118, r119, r120, r121, r122, r123, r124, r125, r126, r127, r128, r129, r130, r131, r132, r133, r134, r135, r136, r137, r138, r139, r140, r141, r142, r143, r144, r145, r146, r147, r148, r149, r150, r151, r152, r153, r154, r155, r156, r157, r158, r159, r160, r161, r162, r163, r164, r165, r166, r167, r168, r169, r170, r171, r172, r173, r174, r175, r176, r177, r178, r179, r180, r181, r182, r183, r184, r185, r186, r187, r188, r189, r190, r191, r192, r193, r194, r195, r196, r197, r198, r199, r200, r201, r202, r203, r204, r205, r206, r207, r208, r209, r210, r211, r212, r213, r214, r215, r216, r217, r218, r219, r220, r221, r222, r223, r224, r225, r226, r227, r228, r229, r230, r231, r232, r233, r234, r235, r236, r237, r238, r239, r240, r241, r242, r243, r244, r245, r246, r247, r248, r249, r250, r251, r252, r253, r254, r255, r256, r257, r258, r259, r260, r261, r262, r263, r264, r265, r266, r267, r268, r269, r270, r271, r272, r273, r274, r275, r276, r277, r278, r279, r280, r281, r282, r283, r284, r285, r286, r287, r288, r289, r290, r291, r292, r293, r294, r295, r296, r297, r298, r299, r300, r301, r302, r303, r304, r305, r306, r307, r308, r309, r310, r311, r312, r313, r314, r315, r316, r317, r318, r319, r320, r321, r322, r323, r324, r325, r326, r327, r328, r329, r330, r331, r332, r333, r334, r335, r336, r337, r338, r339, r340, r341, r342, r343, r344, r345, r346, r347, r348, r349, r350, r351, r352, r353, r354, r355, r356, r357, r358

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    mark#_A(x1) = 795
    U96_A(x1) = 275
    U101_A(x1,x2,x3) = 359
    active#_A(x1) = x1 + 436
    mark_A(x1) = max{34, x1 - 1}
    tt_A = 128
    U102_A(x1,x2,x3) = 359
    isNatKind_A(x1) = 359
    U103_A(x1,x2,x3) = 359
    isNatIListKind_A(x1) = 359
    U104_A(x1,x2,x3) = 359
    U105_A(x1,x2) = 359
    isNat_A(x1) = 359
    U106_A(x1) = 359
    isNatIList_A(x1) = 359
    U11_A(x1,x2) = 359
    U12_A(x1,x2) = 359
    U111_A(x1,x2,x3) = 359
    U112_A(x1,x2,x3) = 359
    U113_A(x1,x2,x3) = 359
    U114_A(x1,x2) = 359
    s_A(x1) = 359
    length_A(x1) = 359
    U13_A(x1) = 359
    isNatList_A(x1) = 359
    U121_A(x1,x2) = 359
    U122_A(x1) = 359
    U131_A(x1,x2,x3,x4) = 359
    U132_A(x1,x2,x3,x4) = 359
    U133_A(x1,x2,x3,x4) = 359
    U134_A(x1,x2,x3,x4) = 359
    U135_A(x1,x2,x3,x4) = 359
    U136_A(x1,x2,x3,x4) = 359
    cons_A(x1,x2) = 69
    take_A(x1,x2) = 359
    U21_A(x1,x2) = 359
    U22_A(x1,x2) = 359
    U23_A(x1) = 359
    U31_A(x1,x2) = 359
    U32_A(x1,x2) = 359
    U33_A(x1) = 359
    U41_A(x1,x2,x3) = 359
    U42_A(x1,x2,x3) = 359
    U43_A(x1,x2,x3) = 359
    U44_A(x1,x2,x3) = 359
    U45_A(x1,x2) = 359
    U46_A(x1) = 359
    U51_A(x1,x2) = 359
    U52_A(x1) = 359
    U61_A(x1,x2) = 359
    U62_A(x1) = 358
    U91_A(x1,x2,x3) = 359
    U92_A(x1,x2,x3) = 359
    U93_A(x1,x2,x3) = 359
    U94_A(x1,x2,x3) = 359
    U95_A(x1,x2) = 359
    U71_A(x1) = 128
    U81_A(x1) = 128
    |0|_A = 108
    active_A(x1) = max{67, x1 - 1}
    zeros_A = 110
    nil_A = 69

The next rules are strictly ordered:

  p106

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(U96(X)) -> mark#(X)
p2: mark#(U101(X1,X2,X3)) -> active#(U101(mark(X1),X2,X3))
p3: active#(U101(tt(),V1,V2)) -> mark#(U102(isNatKind(V1),V1,V2))
p4: mark#(U101(X1,X2,X3)) -> mark#(X1)
p5: mark#(U102(X1,X2,X3)) -> active#(U102(mark(X1),X2,X3))
p6: active#(U102(tt(),V1,V2)) -> mark#(U103(isNatIListKind(V2),V1,V2))
p7: mark#(U102(X1,X2,X3)) -> mark#(X1)
p8: mark#(isNatKind(X)) -> active#(isNatKind(X))
p9: active#(U103(tt(),V1,V2)) -> mark#(U104(isNatIListKind(V2),V1,V2))
p10: mark#(U103(X1,X2,X3)) -> active#(U103(mark(X1),X2,X3))
p11: active#(U104(tt(),V1,V2)) -> mark#(U105(isNat(V1),V2))
p12: mark#(U103(X1,X2,X3)) -> mark#(X1)
p13: mark#(isNatIListKind(X)) -> active#(isNatIListKind(X))
p14: active#(U105(tt(),V2)) -> mark#(U106(isNatIList(V2)))
p15: mark#(U104(X1,X2,X3)) -> active#(U104(mark(X1),X2,X3))
p16: active#(U11(tt(),V1)) -> mark#(U12(isNatIListKind(V1),V1))
p17: mark#(U104(X1,X2,X3)) -> mark#(X1)
p18: mark#(U105(X1,X2)) -> active#(U105(mark(X1),X2))
p19: active#(U111(tt(),L,N)) -> mark#(U112(isNatIListKind(L),L,N))
p20: mark#(U105(X1,X2)) -> mark#(X1)
p21: mark#(isNat(X)) -> active#(isNat(X))
p22: active#(U112(tt(),L,N)) -> mark#(U113(isNat(N),L,N))
p23: mark#(U106(X)) -> mark#(X)
p24: mark#(isNatIList(X)) -> active#(isNatIList(X))
p25: active#(U113(tt(),L,N)) -> mark#(U114(isNatKind(N),L))
p26: mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2))
p27: active#(U114(tt(),L)) -> mark#(s(length(L)))
p28: mark#(U11(X1,X2)) -> mark#(X1)
p29: mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2))
p30: active#(U12(tt(),V1)) -> mark#(U13(isNatList(V1)))
p31: mark#(U12(X1,X2)) -> mark#(X1)
p32: mark#(U111(X1,X2,X3)) -> active#(U111(mark(X1),X2,X3))
p33: active#(U121(tt(),IL)) -> mark#(U122(isNatIListKind(IL)))
p34: mark#(U112(X1,X2,X3)) -> active#(U112(mark(X1),X2,X3))
p35: active#(U131(tt(),IL,M,N)) -> mark#(U132(isNatIListKind(IL),IL,M,N))
p36: mark#(U113(X1,X2,X3)) -> active#(U113(mark(X1),X2,X3))
p37: active#(U132(tt(),IL,M,N)) -> mark#(U133(isNat(M),IL,M,N))
p38: mark#(U114(X1,X2)) -> active#(U114(mark(X1),X2))
p39: active#(U133(tt(),IL,M,N)) -> mark#(U134(isNatKind(M),IL,M,N))
p40: mark#(s(X)) -> active#(s(mark(X)))
p41: active#(U134(tt(),IL,M,N)) -> mark#(U135(isNat(N),IL,M,N))
p42: mark#(s(X)) -> mark#(X)
p43: mark#(length(X)) -> active#(length(mark(X)))
p44: active#(U135(tt(),IL,M,N)) -> mark#(U136(isNatKind(N),IL,M,N))
p45: mark#(length(X)) -> mark#(X)
p46: mark#(U13(X)) -> mark#(X)
p47: mark#(isNatList(X)) -> active#(isNatList(X))
p48: active#(U136(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL)))
p49: mark#(U121(X1,X2)) -> active#(U121(mark(X1),X2))
p50: active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1))
p51: mark#(U122(X)) -> mark#(X)
p52: mark#(U131(X1,X2,X3,X4)) -> active#(U131(mark(X1),X2,X3,X4))
p53: active#(U22(tt(),V1)) -> mark#(U23(isNat(V1)))
p54: mark#(U131(X1,X2,X3,X4)) -> mark#(X1)
p55: mark#(U132(X1,X2,X3,X4)) -> active#(U132(mark(X1),X2,X3,X4))
p56: active#(U31(tt(),V)) -> mark#(U32(isNatIListKind(V),V))
p57: mark#(U133(X1,X2,X3,X4)) -> active#(U133(mark(X1),X2,X3,X4))
p58: active#(U32(tt(),V)) -> mark#(U33(isNatList(V)))
p59: mark#(U134(X1,X2,X3,X4)) -> active#(U134(mark(X1),X2,X3,X4))
p60: active#(U41(tt(),V1,V2)) -> mark#(U42(isNatKind(V1),V1,V2))
p61: mark#(U135(X1,X2,X3,X4)) -> active#(U135(mark(X1),X2,X3,X4))
p62: active#(U42(tt(),V1,V2)) -> mark#(U43(isNatIListKind(V2),V1,V2))
p63: mark#(U136(X1,X2,X3,X4)) -> active#(U136(mark(X1),X2,X3,X4))
p64: active#(U43(tt(),V1,V2)) -> mark#(U44(isNatIListKind(V2),V1,V2))
p65: mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2)))
p66: active#(U44(tt(),V1,V2)) -> mark#(U45(isNat(V1),V2))
p67: mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2))
p68: active#(U45(tt(),V2)) -> mark#(U46(isNatIList(V2)))
p69: mark#(U21(X1,X2)) -> mark#(X1)
p70: mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2))
p71: active#(U51(tt(),V2)) -> mark#(U52(isNatIListKind(V2)))
p72: mark#(U22(X1,X2)) -> mark#(X1)
p73: mark#(U23(X)) -> mark#(X)
p74: mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2))
p75: active#(U61(tt(),V2)) -> mark#(U62(isNatIListKind(V2)))
p76: mark#(U31(X1,X2)) -> mark#(X1)
p77: mark#(U32(X1,X2)) -> active#(U32(mark(X1),X2))
p78: active#(U91(tt(),V1,V2)) -> mark#(U92(isNatKind(V1),V1,V2))
p79: mark#(U32(X1,X2)) -> mark#(X1)
p80: mark#(U33(X)) -> mark#(X)
p81: mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3))
p82: active#(U92(tt(),V1,V2)) -> mark#(U93(isNatIListKind(V2),V1,V2))
p83: mark#(U41(X1,X2,X3)) -> mark#(X1)
p84: mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3))
p85: active#(U93(tt(),V1,V2)) -> mark#(U94(isNatIListKind(V2),V1,V2))
p86: mark#(U42(X1,X2,X3)) -> mark#(X1)
p87: mark#(U43(X1,X2,X3)) -> active#(U43(mark(X1),X2,X3))
p88: active#(U94(tt(),V1,V2)) -> mark#(U95(isNat(V1),V2))
p89: mark#(U43(X1,X2,X3)) -> mark#(X1)
p90: mark#(U44(X1,X2,X3)) -> active#(U44(mark(X1),X2,X3))
p91: active#(U95(tt(),V2)) -> mark#(U96(isNatList(V2)))
p92: mark#(U44(X1,X2,X3)) -> mark#(X1)
p93: mark#(U45(X1,X2)) -> active#(U45(mark(X1),X2))
p94: active#(isNat(length(V1))) -> mark#(U11(isNatIListKind(V1),V1))
p95: mark#(U45(X1,X2)) -> mark#(X1)
p96: mark#(U46(X)) -> active#(U46(mark(X)))
p97: active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1))
p98: mark#(U46(X)) -> mark#(X)
p99: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p100: active#(isNatIList(V)) -> mark#(U31(isNatIListKind(V),V))
p101: mark#(U51(X1,X2)) -> mark#(X1)
p102: mark#(U52(X)) -> mark#(X)
p103: mark#(U61(X1,X2)) -> active#(U61(mark(X1),X2))
p104: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNatKind(V1),V1,V2))
p105: mark#(U61(X1,X2)) -> mark#(X1)
p106: active#(isNatIListKind(cons(V1,V2))) -> mark#(U51(isNatKind(V1),V2))
p107: mark#(U62(X)) -> mark#(X)
p108: mark#(U71(X)) -> mark#(X)
p109: mark#(U81(X)) -> mark#(X)
p110: mark#(U91(X1,X2,X3)) -> active#(U91(mark(X1),X2,X3))
p111: active#(isNatIListKind(take(V1,V2))) -> mark#(U61(isNatKind(V1),V2))
p112: mark#(U91(X1,X2,X3)) -> mark#(X1)
p113: mark#(U92(X1,X2,X3)) -> active#(U92(mark(X1),X2,X3))
p114: active#(isNatKind(length(V1))) -> mark#(U71(isNatIListKind(V1)))
p115: mark#(U92(X1,X2,X3)) -> mark#(X1)
p116: mark#(U93(X1,X2,X3)) -> active#(U93(mark(X1),X2,X3))
p117: active#(isNatKind(s(V1))) -> mark#(U81(isNatKind(V1)))
p118: mark#(U93(X1,X2,X3)) -> mark#(X1)
p119: mark#(U94(X1,X2,X3)) -> active#(U94(mark(X1),X2,X3))
p120: active#(isNatList(cons(V1,V2))) -> mark#(U91(isNatKind(V1),V1,V2))
p121: mark#(U94(X1,X2,X3)) -> mark#(X1)
p122: mark#(U95(X1,X2)) -> active#(U95(mark(X1),X2))
p123: active#(isNatList(take(V1,V2))) -> mark#(U101(isNatKind(V1),V1,V2))
p124: mark#(U95(X1,X2)) -> mark#(X1)
p125: active#(length(cons(N,L))) -> mark#(U111(isNatList(L),L,N))
p126: active#(take(|0|(),IL)) -> mark#(U121(isNatIList(IL),IL))
p127: active#(take(s(M),cons(N,IL))) -> mark#(U131(isNatIList(IL),IL,M,N))

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19, p20, p21, p22, p23, p24, p25, p26, p27, p28, p29, p30, p31, p32, p33, p34, p35, p36, p37, p38, p39, p40, p41, p42, p43, p44, p45, p46, p47, p48, p49, p50, p51, p52, p53, p54, p55, p56, p57, p58, p59, p60, p61, p62, p63, p64, p65, p66, p67, p68, p69, p70, p71, p72, p73, p74, p75, p76, p77, p78, p79, p80, p81, p82, p83, p84, p85, p86, p87, p88, p89, p90, p91, p92, p93, p94, p95, p96, p97, p98, p99, p100, p101, p102, p103, p104, p105, p106, p107, p108, p109, p110, p111, p112, p113, p114, p115, p116, p117, p118, p119, p120, p121, p122, p123, p124, p125, p126, p127}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(U96(X)) -> mark#(X)
p2: mark#(U95(X1,X2)) -> mark#(X1)
p3: mark#(U95(X1,X2)) -> active#(U95(mark(X1),X2))
p4: active#(take(s(M),cons(N,IL))) -> mark#(U131(isNatIList(IL),IL,M,N))
p5: mark#(U94(X1,X2,X3)) -> mark#(X1)
p6: mark#(U94(X1,X2,X3)) -> active#(U94(mark(X1),X2,X3))
p7: active#(take(|0|(),IL)) -> mark#(U121(isNatIList(IL),IL))
p8: mark#(U93(X1,X2,X3)) -> mark#(X1)
p9: mark#(U93(X1,X2,X3)) -> active#(U93(mark(X1),X2,X3))
p10: active#(length(cons(N,L))) -> mark#(U111(isNatList(L),L,N))
p11: mark#(U92(X1,X2,X3)) -> mark#(X1)
p12: mark#(U92(X1,X2,X3)) -> active#(U92(mark(X1),X2,X3))
p13: active#(isNatList(take(V1,V2))) -> mark#(U101(isNatKind(V1),V1,V2))
p14: mark#(U91(X1,X2,X3)) -> mark#(X1)
p15: mark#(U91(X1,X2,X3)) -> active#(U91(mark(X1),X2,X3))
p16: active#(isNatList(cons(V1,V2))) -> mark#(U91(isNatKind(V1),V1,V2))
p17: mark#(U81(X)) -> mark#(X)
p18: mark#(U71(X)) -> mark#(X)
p19: mark#(U62(X)) -> mark#(X)
p20: mark#(U61(X1,X2)) -> mark#(X1)
p21: mark#(U61(X1,X2)) -> active#(U61(mark(X1),X2))
p22: active#(isNatKind(s(V1))) -> mark#(U81(isNatKind(V1)))
p23: mark#(U52(X)) -> mark#(X)
p24: mark#(U51(X1,X2)) -> mark#(X1)
p25: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p26: active#(isNatKind(length(V1))) -> mark#(U71(isNatIListKind(V1)))
p27: mark#(U46(X)) -> mark#(X)
p28: mark#(U46(X)) -> active#(U46(mark(X)))
p29: active#(isNatIListKind(take(V1,V2))) -> mark#(U61(isNatKind(V1),V2))
p30: mark#(U45(X1,X2)) -> mark#(X1)
p31: mark#(U45(X1,X2)) -> active#(U45(mark(X1),X2))
p32: active#(isNatIListKind(cons(V1,V2))) -> mark#(U51(isNatKind(V1),V2))
p33: mark#(U44(X1,X2,X3)) -> mark#(X1)
p34: mark#(U44(X1,X2,X3)) -> active#(U44(mark(X1),X2,X3))
p35: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNatKind(V1),V1,V2))
p36: mark#(U43(X1,X2,X3)) -> mark#(X1)
p37: mark#(U43(X1,X2,X3)) -> active#(U43(mark(X1),X2,X3))
p38: active#(isNatIList(V)) -> mark#(U31(isNatIListKind(V),V))
p39: mark#(U42(X1,X2,X3)) -> mark#(X1)
p40: mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3))
p41: active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1))
p42: mark#(U41(X1,X2,X3)) -> mark#(X1)
p43: mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3))
p44: active#(isNat(length(V1))) -> mark#(U11(isNatIListKind(V1),V1))
p45: mark#(U33(X)) -> mark#(X)
p46: mark#(U32(X1,X2)) -> mark#(X1)
p47: mark#(U32(X1,X2)) -> active#(U32(mark(X1),X2))
p48: active#(U95(tt(),V2)) -> mark#(U96(isNatList(V2)))
p49: mark#(U31(X1,X2)) -> mark#(X1)
p50: mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2))
p51: active#(U94(tt(),V1,V2)) -> mark#(U95(isNat(V1),V2))
p52: mark#(U23(X)) -> mark#(X)
p53: mark#(U22(X1,X2)) -> mark#(X1)
p54: mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2))
p55: active#(U93(tt(),V1,V2)) -> mark#(U94(isNatIListKind(V2),V1,V2))
p56: mark#(U21(X1,X2)) -> mark#(X1)
p57: mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2))
p58: active#(U92(tt(),V1,V2)) -> mark#(U93(isNatIListKind(V2),V1,V2))
p59: mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2)))
p60: active#(U91(tt(),V1,V2)) -> mark#(U92(isNatKind(V1),V1,V2))
p61: mark#(U136(X1,X2,X3,X4)) -> active#(U136(mark(X1),X2,X3,X4))
p62: active#(U61(tt(),V2)) -> mark#(U62(isNatIListKind(V2)))
p63: mark#(U135(X1,X2,X3,X4)) -> active#(U135(mark(X1),X2,X3,X4))
p64: active#(U51(tt(),V2)) -> mark#(U52(isNatIListKind(V2)))
p65: mark#(U134(X1,X2,X3,X4)) -> active#(U134(mark(X1),X2,X3,X4))
p66: active#(U45(tt(),V2)) -> mark#(U46(isNatIList(V2)))
p67: mark#(U133(X1,X2,X3,X4)) -> active#(U133(mark(X1),X2,X3,X4))
p68: active#(U44(tt(),V1,V2)) -> mark#(U45(isNat(V1),V2))
p69: mark#(U132(X1,X2,X3,X4)) -> active#(U132(mark(X1),X2,X3,X4))
p70: active#(U43(tt(),V1,V2)) -> mark#(U44(isNatIListKind(V2),V1,V2))
p71: mark#(U131(X1,X2,X3,X4)) -> mark#(X1)
p72: mark#(U131(X1,X2,X3,X4)) -> active#(U131(mark(X1),X2,X3,X4))
p73: active#(U42(tt(),V1,V2)) -> mark#(U43(isNatIListKind(V2),V1,V2))
p74: mark#(U122(X)) -> mark#(X)
p75: mark#(U121(X1,X2)) -> active#(U121(mark(X1),X2))
p76: active#(U41(tt(),V1,V2)) -> mark#(U42(isNatKind(V1),V1,V2))
p77: mark#(isNatList(X)) -> active#(isNatList(X))
p78: active#(U32(tt(),V)) -> mark#(U33(isNatList(V)))
p79: mark#(U13(X)) -> mark#(X)
p80: mark#(length(X)) -> mark#(X)
p81: mark#(length(X)) -> active#(length(mark(X)))
p82: active#(U31(tt(),V)) -> mark#(U32(isNatIListKind(V),V))
p83: mark#(s(X)) -> mark#(X)
p84: mark#(s(X)) -> active#(s(mark(X)))
p85: active#(U22(tt(),V1)) -> mark#(U23(isNat(V1)))
p86: mark#(U114(X1,X2)) -> active#(U114(mark(X1),X2))
p87: active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1))
p88: mark#(U113(X1,X2,X3)) -> active#(U113(mark(X1),X2,X3))
p89: active#(U136(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL)))
p90: mark#(U112(X1,X2,X3)) -> active#(U112(mark(X1),X2,X3))
p91: active#(U135(tt(),IL,M,N)) -> mark#(U136(isNatKind(N),IL,M,N))
p92: mark#(U111(X1,X2,X3)) -> active#(U111(mark(X1),X2,X3))
p93: active#(U134(tt(),IL,M,N)) -> mark#(U135(isNat(N),IL,M,N))
p94: mark#(U12(X1,X2)) -> mark#(X1)
p95: mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2))
p96: active#(U133(tt(),IL,M,N)) -> mark#(U134(isNatKind(M),IL,M,N))
p97: mark#(U11(X1,X2)) -> mark#(X1)
p98: mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2))
p99: active#(U132(tt(),IL,M,N)) -> mark#(U133(isNat(M),IL,M,N))
p100: mark#(isNatIList(X)) -> active#(isNatIList(X))
p101: active#(U131(tt(),IL,M,N)) -> mark#(U132(isNatIListKind(IL),IL,M,N))
p102: mark#(U106(X)) -> mark#(X)
p103: mark#(isNat(X)) -> active#(isNat(X))
p104: active#(U121(tt(),IL)) -> mark#(U122(isNatIListKind(IL)))
p105: mark#(U105(X1,X2)) -> mark#(X1)
p106: mark#(U105(X1,X2)) -> active#(U105(mark(X1),X2))
p107: active#(U12(tt(),V1)) -> mark#(U13(isNatList(V1)))
p108: mark#(U104(X1,X2,X3)) -> mark#(X1)
p109: mark#(U104(X1,X2,X3)) -> active#(U104(mark(X1),X2,X3))
p110: active#(U114(tt(),L)) -> mark#(s(length(L)))
p111: mark#(isNatIListKind(X)) -> active#(isNatIListKind(X))
p112: active#(U113(tt(),L,N)) -> mark#(U114(isNatKind(N),L))
p113: mark#(U103(X1,X2,X3)) -> mark#(X1)
p114: mark#(U103(X1,X2,X3)) -> active#(U103(mark(X1),X2,X3))
p115: active#(U112(tt(),L,N)) -> mark#(U113(isNat(N),L,N))
p116: mark#(isNatKind(X)) -> active#(isNatKind(X))
p117: active#(U111(tt(),L,N)) -> mark#(U112(isNatIListKind(L),L,N))
p118: mark#(U102(X1,X2,X3)) -> mark#(X1)
p119: mark#(U102(X1,X2,X3)) -> active#(U102(mark(X1),X2,X3))
p120: active#(U11(tt(),V1)) -> mark#(U12(isNatIListKind(V1),V1))
p121: mark#(U101(X1,X2,X3)) -> mark#(X1)
p122: mark#(U101(X1,X2,X3)) -> active#(U101(mark(X1),X2,X3))
p123: active#(U105(tt(),V2)) -> mark#(U106(isNatIList(V2)))
p124: active#(U104(tt(),V1,V2)) -> mark#(U105(isNat(V1),V2))
p125: active#(U103(tt(),V1,V2)) -> mark#(U104(isNatIListKind(V2),V1,V2))
p126: active#(U102(tt(),V1,V2)) -> mark#(U103(isNatIListKind(V2),V1,V2))
p127: active#(U101(tt(),V1,V2)) -> mark#(U102(isNatKind(V1),V1,V2))

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41, r42, r43, r44, r45, r46, r47, r48, r49, r50, r51, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76, r77, r78, r79, r80, r81, r82, r83, r84, r85, r86, r87, r88, r89, r90, r91, r92, r93, r94, r95, r96, r97, r98, r99, r100, r101, r102, r103, r104, r105, r106, r107, r108, r109, r110, r111, r112, r113, r114, r115, r116, r117, r118, r119, r120, r121, r122, r123, r124, r125, r126, r127, r128, r129, r130, r131, r132, r133, r134, r135, r136, r137, r138, r139, r140, r141, r142, r143, r144, r145, r146, r147, r148, r149, r150, r151, r152, r153, r154, r155, r156, r157, r158, r159, r160, r161, r162, r163, r164, r165, r166, r167, r168, r169, r170, r171, r172, r173, r174, r175, r176, r177, r178, r179, r180, r181, r182, r183, r184, r185, r186, r187, r188, r189, r190, r191, r192, r193, r194, r195, r196, r197, r198, r199, r200, r201, r202, r203, r204, r205, r206, r207, r208, r209, r210, r211, r212, r213, r214, r215, r216, r217, r218, r219, r220, r221, r222, r223, r224, r225, r226, r227, r228, r229, r230, r231, r232, r233, r234, r235, r236, r237, r238, r239, r240, r241, r242, r243, r244, r245, r246, r247, r248, r249, r250, r251, r252, r253, r254, r255, r256, r257, r258, r259, r260, r261, r262, r263, r264, r265, r266, r267, r268, r269, r270, r271, r272, r273, r274, r275, r276, r277, r278, r279, r280, r281, r282, r283, r284, r285, r286, r287, r288, r289, r290, r291, r292, r293, r294, r295, r296, r297, r298, r299, r300, r301, r302, r303, r304, r305, r306, r307, r308, r309, r310, r311, r312, r313, r314, r315, r316, r317, r318, r319, r320, r321, r322, r323, r324, r325, r326, r327, r328, r329, r330, r331, r332, r333, r334, r335, r336, r337, r338, r339, r340, r341, r342, r343, r344, r345, r346, r347, r348, r349, r350, r351, r352, r353, r354, r355, r356, r357, r358

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    mark#_A(x1) = 125
    U96_A(x1) = 83
    U95_A(x1,x2) = 103
    active#_A(x1) = x1 + 22
    mark_A(x1) = max{68, x1 - 6}
    take_A(x1,x2) = 103
    s_A(x1) = 42
    cons_A(x1,x2) = 70
    U131_A(x1,x2,x3,x4) = 103
    isNatIList_A(x1) = 103
    U94_A(x1,x2,x3) = 103
    |0|_A = 75
    U121_A(x1,x2) = 103
    U93_A(x1,x2,x3) = 103
    length_A(x1) = 103
    U111_A(x1,x2,x3) = 103
    isNatList_A(x1) = 103
    U92_A(x1,x2,x3) = 103
    U101_A(x1,x2,x3) = 103
    isNatKind_A(x1) = 103
    U91_A(x1,x2,x3) = 103
    U81_A(x1) = 103
    U71_A(x1) = 103
    U62_A(x1) = 102
    U61_A(x1,x2) = 103
    U52_A(x1) = 102
    U51_A(x1,x2) = 103
    isNatIListKind_A(x1) = 103
    U46_A(x1) = 103
    U45_A(x1,x2) = 103
    U44_A(x1,x2,x3) = 103
    U41_A(x1,x2,x3) = 103
    U43_A(x1,x2,x3) = 103
    U31_A(x1,x2) = 103
    U42_A(x1,x2,x3) = 103
    isNat_A(x1) = 103
    U21_A(x1,x2) = 103
    U11_A(x1,x2) = 103
    U33_A(x1) = 103
    U32_A(x1,x2) = 103
    tt_A = 75
    U23_A(x1) = 103
    U22_A(x1,x2) = 103
    U136_A(x1,x2,x3,x4) = 103
    U135_A(x1,x2,x3,x4) = 103
    U134_A(x1,x2,x3,x4) = 103
    U133_A(x1,x2,x3,x4) = 103
    U132_A(x1,x2,x3,x4) = 103
    U122_A(x1) = 103
    U13_A(x1) = 76
    U114_A(x1,x2) = 103
    U113_A(x1,x2,x3) = 103
    U112_A(x1,x2,x3) = 103
    U12_A(x1,x2) = 103
    U106_A(x1) = 76
    U105_A(x1,x2) = 103
    U104_A(x1,x2,x3) = 103
    U103_A(x1,x2,x3) = 103
    U102_A(x1,x2,x3) = 103
    active_A(x1) = max{68, x1 - 6}
    zeros_A = 0
    nil_A = 103

The next rules are strictly ordered:

  p84

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(U96(X)) -> mark#(X)
p2: mark#(U95(X1,X2)) -> mark#(X1)
p3: mark#(U95(X1,X2)) -> active#(U95(mark(X1),X2))
p4: active#(take(s(M),cons(N,IL))) -> mark#(U131(isNatIList(IL),IL,M,N))
p5: mark#(U94(X1,X2,X3)) -> mark#(X1)
p6: mark#(U94(X1,X2,X3)) -> active#(U94(mark(X1),X2,X3))
p7: active#(take(|0|(),IL)) -> mark#(U121(isNatIList(IL),IL))
p8: mark#(U93(X1,X2,X3)) -> mark#(X1)
p9: mark#(U93(X1,X2,X3)) -> active#(U93(mark(X1),X2,X3))
p10: active#(length(cons(N,L))) -> mark#(U111(isNatList(L),L,N))
p11: mark#(U92(X1,X2,X3)) -> mark#(X1)
p12: mark#(U92(X1,X2,X3)) -> active#(U92(mark(X1),X2,X3))
p13: active#(isNatList(take(V1,V2))) -> mark#(U101(isNatKind(V1),V1,V2))
p14: mark#(U91(X1,X2,X3)) -> mark#(X1)
p15: mark#(U91(X1,X2,X3)) -> active#(U91(mark(X1),X2,X3))
p16: active#(isNatList(cons(V1,V2))) -> mark#(U91(isNatKind(V1),V1,V2))
p17: mark#(U81(X)) -> mark#(X)
p18: mark#(U71(X)) -> mark#(X)
p19: mark#(U62(X)) -> mark#(X)
p20: mark#(U61(X1,X2)) -> mark#(X1)
p21: mark#(U61(X1,X2)) -> active#(U61(mark(X1),X2))
p22: active#(isNatKind(s(V1))) -> mark#(U81(isNatKind(V1)))
p23: mark#(U52(X)) -> mark#(X)
p24: mark#(U51(X1,X2)) -> mark#(X1)
p25: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p26: active#(isNatKind(length(V1))) -> mark#(U71(isNatIListKind(V1)))
p27: mark#(U46(X)) -> mark#(X)
p28: mark#(U46(X)) -> active#(U46(mark(X)))
p29: active#(isNatIListKind(take(V1,V2))) -> mark#(U61(isNatKind(V1),V2))
p30: mark#(U45(X1,X2)) -> mark#(X1)
p31: mark#(U45(X1,X2)) -> active#(U45(mark(X1),X2))
p32: active#(isNatIListKind(cons(V1,V2))) -> mark#(U51(isNatKind(V1),V2))
p33: mark#(U44(X1,X2,X3)) -> mark#(X1)
p34: mark#(U44(X1,X2,X3)) -> active#(U44(mark(X1),X2,X3))
p35: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNatKind(V1),V1,V2))
p36: mark#(U43(X1,X2,X3)) -> mark#(X1)
p37: mark#(U43(X1,X2,X3)) -> active#(U43(mark(X1),X2,X3))
p38: active#(isNatIList(V)) -> mark#(U31(isNatIListKind(V),V))
p39: mark#(U42(X1,X2,X3)) -> mark#(X1)
p40: mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3))
p41: active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1))
p42: mark#(U41(X1,X2,X3)) -> mark#(X1)
p43: mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3))
p44: active#(isNat(length(V1))) -> mark#(U11(isNatIListKind(V1),V1))
p45: mark#(U33(X)) -> mark#(X)
p46: mark#(U32(X1,X2)) -> mark#(X1)
p47: mark#(U32(X1,X2)) -> active#(U32(mark(X1),X2))
p48: active#(U95(tt(),V2)) -> mark#(U96(isNatList(V2)))
p49: mark#(U31(X1,X2)) -> mark#(X1)
p50: mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2))
p51: active#(U94(tt(),V1,V2)) -> mark#(U95(isNat(V1),V2))
p52: mark#(U23(X)) -> mark#(X)
p53: mark#(U22(X1,X2)) -> mark#(X1)
p54: mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2))
p55: active#(U93(tt(),V1,V2)) -> mark#(U94(isNatIListKind(V2),V1,V2))
p56: mark#(U21(X1,X2)) -> mark#(X1)
p57: mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2))
p58: active#(U92(tt(),V1,V2)) -> mark#(U93(isNatIListKind(V2),V1,V2))
p59: mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2)))
p60: active#(U91(tt(),V1,V2)) -> mark#(U92(isNatKind(V1),V1,V2))
p61: mark#(U136(X1,X2,X3,X4)) -> active#(U136(mark(X1),X2,X3,X4))
p62: active#(U61(tt(),V2)) -> mark#(U62(isNatIListKind(V2)))
p63: mark#(U135(X1,X2,X3,X4)) -> active#(U135(mark(X1),X2,X3,X4))
p64: active#(U51(tt(),V2)) -> mark#(U52(isNatIListKind(V2)))
p65: mark#(U134(X1,X2,X3,X4)) -> active#(U134(mark(X1),X2,X3,X4))
p66: active#(U45(tt(),V2)) -> mark#(U46(isNatIList(V2)))
p67: mark#(U133(X1,X2,X3,X4)) -> active#(U133(mark(X1),X2,X3,X4))
p68: active#(U44(tt(),V1,V2)) -> mark#(U45(isNat(V1),V2))
p69: mark#(U132(X1,X2,X3,X4)) -> active#(U132(mark(X1),X2,X3,X4))
p70: active#(U43(tt(),V1,V2)) -> mark#(U44(isNatIListKind(V2),V1,V2))
p71: mark#(U131(X1,X2,X3,X4)) -> mark#(X1)
p72: mark#(U131(X1,X2,X3,X4)) -> active#(U131(mark(X1),X2,X3,X4))
p73: active#(U42(tt(),V1,V2)) -> mark#(U43(isNatIListKind(V2),V1,V2))
p74: mark#(U122(X)) -> mark#(X)
p75: mark#(U121(X1,X2)) -> active#(U121(mark(X1),X2))
p76: active#(U41(tt(),V1,V2)) -> mark#(U42(isNatKind(V1),V1,V2))
p77: mark#(isNatList(X)) -> active#(isNatList(X))
p78: active#(U32(tt(),V)) -> mark#(U33(isNatList(V)))
p79: mark#(U13(X)) -> mark#(X)
p80: mark#(length(X)) -> mark#(X)
p81: mark#(length(X)) -> active#(length(mark(X)))
p82: active#(U31(tt(),V)) -> mark#(U32(isNatIListKind(V),V))
p83: mark#(s(X)) -> mark#(X)
p84: active#(U22(tt(),V1)) -> mark#(U23(isNat(V1)))
p85: mark#(U114(X1,X2)) -> active#(U114(mark(X1),X2))
p86: active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1))
p87: mark#(U113(X1,X2,X3)) -> active#(U113(mark(X1),X2,X3))
p88: active#(U136(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL)))
p89: mark#(U112(X1,X2,X3)) -> active#(U112(mark(X1),X2,X3))
p90: active#(U135(tt(),IL,M,N)) -> mark#(U136(isNatKind(N),IL,M,N))
p91: mark#(U111(X1,X2,X3)) -> active#(U111(mark(X1),X2,X3))
p92: active#(U134(tt(),IL,M,N)) -> mark#(U135(isNat(N),IL,M,N))
p93: mark#(U12(X1,X2)) -> mark#(X1)
p94: mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2))
p95: active#(U133(tt(),IL,M,N)) -> mark#(U134(isNatKind(M),IL,M,N))
p96: mark#(U11(X1,X2)) -> mark#(X1)
p97: mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2))
p98: active#(U132(tt(),IL,M,N)) -> mark#(U133(isNat(M),IL,M,N))
p99: mark#(isNatIList(X)) -> active#(isNatIList(X))
p100: active#(U131(tt(),IL,M,N)) -> mark#(U132(isNatIListKind(IL),IL,M,N))
p101: mark#(U106(X)) -> mark#(X)
p102: mark#(isNat(X)) -> active#(isNat(X))
p103: active#(U121(tt(),IL)) -> mark#(U122(isNatIListKind(IL)))
p104: mark#(U105(X1,X2)) -> mark#(X1)
p105: mark#(U105(X1,X2)) -> active#(U105(mark(X1),X2))
p106: active#(U12(tt(),V1)) -> mark#(U13(isNatList(V1)))
p107: mark#(U104(X1,X2,X3)) -> mark#(X1)
p108: mark#(U104(X1,X2,X3)) -> active#(U104(mark(X1),X2,X3))
p109: active#(U114(tt(),L)) -> mark#(s(length(L)))
p110: mark#(isNatIListKind(X)) -> active#(isNatIListKind(X))
p111: active#(U113(tt(),L,N)) -> mark#(U114(isNatKind(N),L))
p112: mark#(U103(X1,X2,X3)) -> mark#(X1)
p113: mark#(U103(X1,X2,X3)) -> active#(U103(mark(X1),X2,X3))
p114: active#(U112(tt(),L,N)) -> mark#(U113(isNat(N),L,N))
p115: mark#(isNatKind(X)) -> active#(isNatKind(X))
p116: active#(U111(tt(),L,N)) -> mark#(U112(isNatIListKind(L),L,N))
p117: mark#(U102(X1,X2,X3)) -> mark#(X1)
p118: mark#(U102(X1,X2,X3)) -> active#(U102(mark(X1),X2,X3))
p119: active#(U11(tt(),V1)) -> mark#(U12(isNatIListKind(V1),V1))
p120: mark#(U101(X1,X2,X3)) -> mark#(X1)
p121: mark#(U101(X1,X2,X3)) -> active#(U101(mark(X1),X2,X3))
p122: active#(U105(tt(),V2)) -> mark#(U106(isNatIList(V2)))
p123: active#(U104(tt(),V1,V2)) -> mark#(U105(isNat(V1),V2))
p124: active#(U103(tt(),V1,V2)) -> mark#(U104(isNatIListKind(V2),V1,V2))
p125: active#(U102(tt(),V1,V2)) -> mark#(U103(isNatIListKind(V2),V1,V2))
p126: active#(U101(tt(),V1,V2)) -> mark#(U102(isNatKind(V1),V1,V2))

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19, p20, p21, p22, p23, p24, p25, p26, p27, p28, p29, p30, p31, p32, p33, p34, p35, p36, p37, p38, p39, p40, p41, p42, p43, p44, p45, p46, p47, p48, p49, p50, p51, p52, p53, p54, p55, p56, p57, p58, p59, p60, p61, p62, p63, p64, p65, p66, p67, p68, p69, p70, p71, p72, p73, p74, p75, p76, p77, p78, p79, p80, p81, p82, p83, p84, p85, p86, p87, p88, p89, p90, p91, p92, p93, p94, p95, p96, p97, p98, p99, p100, p101, p102, p103, p104, p105, p106, p107, p108, p109, p110, p111, p112, p113, p114, p115, p116, p117, p118, p119, p120, p121, p122, p123, p124, p125, p126}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(U96(X)) -> mark#(X)
p2: mark#(U101(X1,X2,X3)) -> active#(U101(mark(X1),X2,X3))
p3: active#(U101(tt(),V1,V2)) -> mark#(U102(isNatKind(V1),V1,V2))
p4: mark#(U101(X1,X2,X3)) -> mark#(X1)
p5: mark#(U102(X1,X2,X3)) -> active#(U102(mark(X1),X2,X3))
p6: active#(U102(tt(),V1,V2)) -> mark#(U103(isNatIListKind(V2),V1,V2))
p7: mark#(U102(X1,X2,X3)) -> mark#(X1)
p8: mark#(isNatKind(X)) -> active#(isNatKind(X))
p9: active#(U103(tt(),V1,V2)) -> mark#(U104(isNatIListKind(V2),V1,V2))
p10: mark#(U103(X1,X2,X3)) -> active#(U103(mark(X1),X2,X3))
p11: active#(U104(tt(),V1,V2)) -> mark#(U105(isNat(V1),V2))
p12: mark#(U103(X1,X2,X3)) -> mark#(X1)
p13: mark#(isNatIListKind(X)) -> active#(isNatIListKind(X))
p14: active#(U105(tt(),V2)) -> mark#(U106(isNatIList(V2)))
p15: mark#(U104(X1,X2,X3)) -> active#(U104(mark(X1),X2,X3))
p16: active#(U11(tt(),V1)) -> mark#(U12(isNatIListKind(V1),V1))
p17: mark#(U104(X1,X2,X3)) -> mark#(X1)
p18: mark#(U105(X1,X2)) -> active#(U105(mark(X1),X2))
p19: active#(U111(tt(),L,N)) -> mark#(U112(isNatIListKind(L),L,N))
p20: mark#(U105(X1,X2)) -> mark#(X1)
p21: mark#(isNat(X)) -> active#(isNat(X))
p22: active#(U112(tt(),L,N)) -> mark#(U113(isNat(N),L,N))
p23: mark#(U106(X)) -> mark#(X)
p24: mark#(isNatIList(X)) -> active#(isNatIList(X))
p25: active#(U113(tt(),L,N)) -> mark#(U114(isNatKind(N),L))
p26: mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2))
p27: active#(U114(tt(),L)) -> mark#(s(length(L)))
p28: mark#(U11(X1,X2)) -> mark#(X1)
p29: mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2))
p30: active#(U12(tt(),V1)) -> mark#(U13(isNatList(V1)))
p31: mark#(U12(X1,X2)) -> mark#(X1)
p32: mark#(U111(X1,X2,X3)) -> active#(U111(mark(X1),X2,X3))
p33: active#(U121(tt(),IL)) -> mark#(U122(isNatIListKind(IL)))
p34: mark#(U112(X1,X2,X3)) -> active#(U112(mark(X1),X2,X3))
p35: active#(U131(tt(),IL,M,N)) -> mark#(U132(isNatIListKind(IL),IL,M,N))
p36: mark#(U113(X1,X2,X3)) -> active#(U113(mark(X1),X2,X3))
p37: active#(U132(tt(),IL,M,N)) -> mark#(U133(isNat(M),IL,M,N))
p38: mark#(U114(X1,X2)) -> active#(U114(mark(X1),X2))
p39: active#(U133(tt(),IL,M,N)) -> mark#(U134(isNatKind(M),IL,M,N))
p40: mark#(s(X)) -> mark#(X)
p41: mark#(length(X)) -> active#(length(mark(X)))
p42: active#(U134(tt(),IL,M,N)) -> mark#(U135(isNat(N),IL,M,N))
p43: mark#(length(X)) -> mark#(X)
p44: mark#(U13(X)) -> mark#(X)
p45: mark#(isNatList(X)) -> active#(isNatList(X))
p46: active#(U135(tt(),IL,M,N)) -> mark#(U136(isNatKind(N),IL,M,N))
p47: mark#(U121(X1,X2)) -> active#(U121(mark(X1),X2))
p48: active#(U136(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL)))
p49: mark#(U122(X)) -> mark#(X)
p50: mark#(U131(X1,X2,X3,X4)) -> active#(U131(mark(X1),X2,X3,X4))
p51: active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1))
p52: mark#(U131(X1,X2,X3,X4)) -> mark#(X1)
p53: mark#(U132(X1,X2,X3,X4)) -> active#(U132(mark(X1),X2,X3,X4))
p54: active#(U22(tt(),V1)) -> mark#(U23(isNat(V1)))
p55: mark#(U133(X1,X2,X3,X4)) -> active#(U133(mark(X1),X2,X3,X4))
p56: active#(U31(tt(),V)) -> mark#(U32(isNatIListKind(V),V))
p57: mark#(U134(X1,X2,X3,X4)) -> active#(U134(mark(X1),X2,X3,X4))
p58: active#(U32(tt(),V)) -> mark#(U33(isNatList(V)))
p59: mark#(U135(X1,X2,X3,X4)) -> active#(U135(mark(X1),X2,X3,X4))
p60: active#(U41(tt(),V1,V2)) -> mark#(U42(isNatKind(V1),V1,V2))
p61: mark#(U136(X1,X2,X3,X4)) -> active#(U136(mark(X1),X2,X3,X4))
p62: active#(U42(tt(),V1,V2)) -> mark#(U43(isNatIListKind(V2),V1,V2))
p63: mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2)))
p64: active#(U43(tt(),V1,V2)) -> mark#(U44(isNatIListKind(V2),V1,V2))
p65: mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2))
p66: active#(U44(tt(),V1,V2)) -> mark#(U45(isNat(V1),V2))
p67: mark#(U21(X1,X2)) -> mark#(X1)
p68: mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2))
p69: active#(U45(tt(),V2)) -> mark#(U46(isNatIList(V2)))
p70: mark#(U22(X1,X2)) -> mark#(X1)
p71: mark#(U23(X)) -> mark#(X)
p72: mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2))
p73: active#(U51(tt(),V2)) -> mark#(U52(isNatIListKind(V2)))
p74: mark#(U31(X1,X2)) -> mark#(X1)
p75: mark#(U32(X1,X2)) -> active#(U32(mark(X1),X2))
p76: active#(U61(tt(),V2)) -> mark#(U62(isNatIListKind(V2)))
p77: mark#(U32(X1,X2)) -> mark#(X1)
p78: mark#(U33(X)) -> mark#(X)
p79: mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3))
p80: active#(U91(tt(),V1,V2)) -> mark#(U92(isNatKind(V1),V1,V2))
p81: mark#(U41(X1,X2,X3)) -> mark#(X1)
p82: mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3))
p83: active#(U92(tt(),V1,V2)) -> mark#(U93(isNatIListKind(V2),V1,V2))
p84: mark#(U42(X1,X2,X3)) -> mark#(X1)
p85: mark#(U43(X1,X2,X3)) -> active#(U43(mark(X1),X2,X3))
p86: active#(U93(tt(),V1,V2)) -> mark#(U94(isNatIListKind(V2),V1,V2))
p87: mark#(U43(X1,X2,X3)) -> mark#(X1)
p88: mark#(U44(X1,X2,X3)) -> active#(U44(mark(X1),X2,X3))
p89: active#(U94(tt(),V1,V2)) -> mark#(U95(isNat(V1),V2))
p90: mark#(U44(X1,X2,X3)) -> mark#(X1)
p91: mark#(U45(X1,X2)) -> active#(U45(mark(X1),X2))
p92: active#(U95(tt(),V2)) -> mark#(U96(isNatList(V2)))
p93: mark#(U45(X1,X2)) -> mark#(X1)
p94: mark#(U46(X)) -> active#(U46(mark(X)))
p95: active#(isNat(length(V1))) -> mark#(U11(isNatIListKind(V1),V1))
p96: mark#(U46(X)) -> mark#(X)
p97: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p98: active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1))
p99: mark#(U51(X1,X2)) -> mark#(X1)
p100: mark#(U52(X)) -> mark#(X)
p101: mark#(U61(X1,X2)) -> active#(U61(mark(X1),X2))
p102: active#(isNatIList(V)) -> mark#(U31(isNatIListKind(V),V))
p103: mark#(U61(X1,X2)) -> mark#(X1)
p104: mark#(U62(X)) -> mark#(X)
p105: mark#(U71(X)) -> mark#(X)
p106: mark#(U81(X)) -> mark#(X)
p107: mark#(U91(X1,X2,X3)) -> active#(U91(mark(X1),X2,X3))
p108: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNatKind(V1),V1,V2))
p109: mark#(U91(X1,X2,X3)) -> mark#(X1)
p110: mark#(U92(X1,X2,X3)) -> active#(U92(mark(X1),X2,X3))
p111: active#(isNatIListKind(cons(V1,V2))) -> mark#(U51(isNatKind(V1),V2))
p112: mark#(U92(X1,X2,X3)) -> mark#(X1)
p113: mark#(U93(X1,X2,X3)) -> active#(U93(mark(X1),X2,X3))
p114: active#(isNatIListKind(take(V1,V2))) -> mark#(U61(isNatKind(V1),V2))
p115: mark#(U93(X1,X2,X3)) -> mark#(X1)
p116: mark#(U94(X1,X2,X3)) -> active#(U94(mark(X1),X2,X3))
p117: active#(isNatKind(length(V1))) -> mark#(U71(isNatIListKind(V1)))
p118: mark#(U94(X1,X2,X3)) -> mark#(X1)
p119: mark#(U95(X1,X2)) -> active#(U95(mark(X1),X2))
p120: active#(isNatKind(s(V1))) -> mark#(U81(isNatKind(V1)))
p121: mark#(U95(X1,X2)) -> mark#(X1)
p122: active#(isNatList(cons(V1,V2))) -> mark#(U91(isNatKind(V1),V1,V2))
p123: active#(isNatList(take(V1,V2))) -> mark#(U101(isNatKind(V1),V1,V2))
p124: active#(length(cons(N,L))) -> mark#(U111(isNatList(L),L,N))
p125: active#(take(|0|(),IL)) -> mark#(U121(isNatIList(IL),IL))
p126: active#(take(s(M),cons(N,IL))) -> mark#(U131(isNatIList(IL),IL,M,N))

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41, r42, r43, r44, r45, r46, r47, r48, r49, r50, r51, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76, r77, r78, r79, r80, r81, r82, r83, r84, r85, r86, r87, r88, r89, r90, r91, r92, r93, r94, r95, r96, r97, r98, r99, r100, r101, r102, r103, r104, r105, r106, r107, r108, r109, r110, r111, r112, r113, r114, r115, r116, r117, r118, r119, r120, r121, r122, r123, r124, r125, r126, r127, r128, r129, r130, r131, r132, r133, r134, r135, r136, r137, r138, r139, r140, r141, r142, r143, r144, r145, r146, r147, r148, r149, r150, r151, r152, r153, r154, r155, r156, r157, r158, r159, r160, r161, r162, r163, r164, r165, r166, r167, r168, r169, r170, r171, r172, r173, r174, r175, r176, r177, r178, r179, r180, r181, r182, r183, r184, r185, r186, r187, r188, r189, r190, r191, r192, r193, r194, r195, r196, r197, r198, r199, r200, r201, r202, r203, r204, r205, r206, r207, r208, r209, r210, r211, r212, r213, r214, r215, r216, r217, r218, r219, r220, r221, r222, r223, r224, r225, r226, r227, r228, r229, r230, r231, r232, r233, r234, r235, r236, r237, r238, r239, r240, r241, r242, r243, r244, r245, r246, r247, r248, r249, r250, r251, r252, r253, r254, r255, r256, r257, r258, r259, r260, r261, r262, r263, r264, r265, r266, r267, r268, r269, r270, r271, r272, r273, r274, r275, r276, r277, r278, r279, r280, r281, r282, r283, r284, r285, r286, r287, r288, r289, r290, r291, r292, r293, r294, r295, r296, r297, r298, r299, r300, r301, r302, r303, r304, r305, r306, r307, r308, r309, r310, r311, r312, r313, r314, r315, r316, r317, r318, r319, r320, r321, r322, r323, r324, r325, r326, r327, r328, r329, r330, r331, r332, r333, r334, r335, r336, r337, r338, r339, r340, r341, r342, r343, r344, r345, r346, r347, r348, r349, r350, r351, r352, r353, r354, r355, r356, r357, r358

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    mark#_A(x1) = 147
    U96_A(x1) = 26
    U101_A(x1,x2,x3) = 156
    active#_A(x1) = max{0, x1 - 9}
    mark_A(x1) = max{4, x1 - 18}
    tt_A = 23
    U102_A(x1,x2,x3) = 156
    isNatKind_A(x1) = 156
    U103_A(x1,x2,x3) = 156
    isNatIListKind_A(x1) = 156
    U104_A(x1,x2,x3) = 156
    U105_A(x1,x2) = 156
    isNat_A(x1) = 156
    U106_A(x1) = 26
    isNatIList_A(x1) = 156
    U11_A(x1,x2) = 156
    U12_A(x1,x2) = 156
    U111_A(x1,x2,x3) = 156
    U112_A(x1,x2,x3) = 156
    U113_A(x1,x2,x3) = 156
    U114_A(x1,x2) = 156
    s_A(x1) = 26
    length_A(x1) = 156
    U13_A(x1) = 58
    isNatList_A(x1) = 156
    U121_A(x1,x2) = 156
    U122_A(x1) = 50
    U131_A(x1,x2,x3,x4) = 156
    U132_A(x1,x2,x3,x4) = 156
    U133_A(x1,x2,x3,x4) = 156
    U134_A(x1,x2,x3,x4) = 156
    U135_A(x1,x2,x3,x4) = 156
    U136_A(x1,x2,x3,x4) = 156
    cons_A(x1,x2) = 24
    take_A(x1,x2) = 156
    U21_A(x1,x2) = 156
    U22_A(x1,x2) = 156
    U23_A(x1) = 156
    U31_A(x1,x2) = 156
    U32_A(x1,x2) = 156
    U33_A(x1) = 156
    U41_A(x1,x2,x3) = 156
    U42_A(x1,x2,x3) = 156
    U43_A(x1,x2,x3) = 156
    U44_A(x1,x2,x3) = 156
    U45_A(x1,x2) = 156
    U46_A(x1) = 23
    U51_A(x1,x2) = 156
    U52_A(x1) = 24
    U61_A(x1,x2) = 156
    U62_A(x1) = 156
    U91_A(x1,x2,x3) = 156
    U92_A(x1,x2,x3) = 156
    U93_A(x1,x2,x3) = 156
    U94_A(x1,x2,x3) = 156
    U95_A(x1,x2) = 156
    U71_A(x1) = 156
    U81_A(x1) = 24
    |0|_A = 24
    active_A(x1) = max{5, x1 - 18}
    zeros_A = 24
    nil_A = 24

The next rules are strictly ordered:

  p94

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(U96(X)) -> mark#(X)
p2: mark#(U101(X1,X2,X3)) -> active#(U101(mark(X1),X2,X3))
p3: active#(U101(tt(),V1,V2)) -> mark#(U102(isNatKind(V1),V1,V2))
p4: mark#(U101(X1,X2,X3)) -> mark#(X1)
p5: mark#(U102(X1,X2,X3)) -> active#(U102(mark(X1),X2,X3))
p6: active#(U102(tt(),V1,V2)) -> mark#(U103(isNatIListKind(V2),V1,V2))
p7: mark#(U102(X1,X2,X3)) -> mark#(X1)
p8: mark#(isNatKind(X)) -> active#(isNatKind(X))
p9: active#(U103(tt(),V1,V2)) -> mark#(U104(isNatIListKind(V2),V1,V2))
p10: mark#(U103(X1,X2,X3)) -> active#(U103(mark(X1),X2,X3))
p11: active#(U104(tt(),V1,V2)) -> mark#(U105(isNat(V1),V2))
p12: mark#(U103(X1,X2,X3)) -> mark#(X1)
p13: mark#(isNatIListKind(X)) -> active#(isNatIListKind(X))
p14: active#(U105(tt(),V2)) -> mark#(U106(isNatIList(V2)))
p15: mark#(U104(X1,X2,X3)) -> active#(U104(mark(X1),X2,X3))
p16: active#(U11(tt(),V1)) -> mark#(U12(isNatIListKind(V1),V1))
p17: mark#(U104(X1,X2,X3)) -> mark#(X1)
p18: mark#(U105(X1,X2)) -> active#(U105(mark(X1),X2))
p19: active#(U111(tt(),L,N)) -> mark#(U112(isNatIListKind(L),L,N))
p20: mark#(U105(X1,X2)) -> mark#(X1)
p21: mark#(isNat(X)) -> active#(isNat(X))
p22: active#(U112(tt(),L,N)) -> mark#(U113(isNat(N),L,N))
p23: mark#(U106(X)) -> mark#(X)
p24: mark#(isNatIList(X)) -> active#(isNatIList(X))
p25: active#(U113(tt(),L,N)) -> mark#(U114(isNatKind(N),L))
p26: mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2))
p27: active#(U114(tt(),L)) -> mark#(s(length(L)))
p28: mark#(U11(X1,X2)) -> mark#(X1)
p29: mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2))
p30: active#(U12(tt(),V1)) -> mark#(U13(isNatList(V1)))
p31: mark#(U12(X1,X2)) -> mark#(X1)
p32: mark#(U111(X1,X2,X3)) -> active#(U111(mark(X1),X2,X3))
p33: active#(U121(tt(),IL)) -> mark#(U122(isNatIListKind(IL)))
p34: mark#(U112(X1,X2,X3)) -> active#(U112(mark(X1),X2,X3))
p35: active#(U131(tt(),IL,M,N)) -> mark#(U132(isNatIListKind(IL),IL,M,N))
p36: mark#(U113(X1,X2,X3)) -> active#(U113(mark(X1),X2,X3))
p37: active#(U132(tt(),IL,M,N)) -> mark#(U133(isNat(M),IL,M,N))
p38: mark#(U114(X1,X2)) -> active#(U114(mark(X1),X2))
p39: active#(U133(tt(),IL,M,N)) -> mark#(U134(isNatKind(M),IL,M,N))
p40: mark#(s(X)) -> mark#(X)
p41: mark#(length(X)) -> active#(length(mark(X)))
p42: active#(U134(tt(),IL,M,N)) -> mark#(U135(isNat(N),IL,M,N))
p43: mark#(length(X)) -> mark#(X)
p44: mark#(U13(X)) -> mark#(X)
p45: mark#(isNatList(X)) -> active#(isNatList(X))
p46: active#(U135(tt(),IL,M,N)) -> mark#(U136(isNatKind(N),IL,M,N))
p47: mark#(U121(X1,X2)) -> active#(U121(mark(X1),X2))
p48: active#(U136(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL)))
p49: mark#(U122(X)) -> mark#(X)
p50: mark#(U131(X1,X2,X3,X4)) -> active#(U131(mark(X1),X2,X3,X4))
p51: active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1))
p52: mark#(U131(X1,X2,X3,X4)) -> mark#(X1)
p53: mark#(U132(X1,X2,X3,X4)) -> active#(U132(mark(X1),X2,X3,X4))
p54: active#(U22(tt(),V1)) -> mark#(U23(isNat(V1)))
p55: mark#(U133(X1,X2,X3,X4)) -> active#(U133(mark(X1),X2,X3,X4))
p56: active#(U31(tt(),V)) -> mark#(U32(isNatIListKind(V),V))
p57: mark#(U134(X1,X2,X3,X4)) -> active#(U134(mark(X1),X2,X3,X4))
p58: active#(U32(tt(),V)) -> mark#(U33(isNatList(V)))
p59: mark#(U135(X1,X2,X3,X4)) -> active#(U135(mark(X1),X2,X3,X4))
p60: active#(U41(tt(),V1,V2)) -> mark#(U42(isNatKind(V1),V1,V2))
p61: mark#(U136(X1,X2,X3,X4)) -> active#(U136(mark(X1),X2,X3,X4))
p62: active#(U42(tt(),V1,V2)) -> mark#(U43(isNatIListKind(V2),V1,V2))
p63: mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2)))
p64: active#(U43(tt(),V1,V2)) -> mark#(U44(isNatIListKind(V2),V1,V2))
p65: mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2))
p66: active#(U44(tt(),V1,V2)) -> mark#(U45(isNat(V1),V2))
p67: mark#(U21(X1,X2)) -> mark#(X1)
p68: mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2))
p69: active#(U45(tt(),V2)) -> mark#(U46(isNatIList(V2)))
p70: mark#(U22(X1,X2)) -> mark#(X1)
p71: mark#(U23(X)) -> mark#(X)
p72: mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2))
p73: active#(U51(tt(),V2)) -> mark#(U52(isNatIListKind(V2)))
p74: mark#(U31(X1,X2)) -> mark#(X1)
p75: mark#(U32(X1,X2)) -> active#(U32(mark(X1),X2))
p76: active#(U61(tt(),V2)) -> mark#(U62(isNatIListKind(V2)))
p77: mark#(U32(X1,X2)) -> mark#(X1)
p78: mark#(U33(X)) -> mark#(X)
p79: mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3))
p80: active#(U91(tt(),V1,V2)) -> mark#(U92(isNatKind(V1),V1,V2))
p81: mark#(U41(X1,X2,X3)) -> mark#(X1)
p82: mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3))
p83: active#(U92(tt(),V1,V2)) -> mark#(U93(isNatIListKind(V2),V1,V2))
p84: mark#(U42(X1,X2,X3)) -> mark#(X1)
p85: mark#(U43(X1,X2,X3)) -> active#(U43(mark(X1),X2,X3))
p86: active#(U93(tt(),V1,V2)) -> mark#(U94(isNatIListKind(V2),V1,V2))
p87: mark#(U43(X1,X2,X3)) -> mark#(X1)
p88: mark#(U44(X1,X2,X3)) -> active#(U44(mark(X1),X2,X3))
p89: active#(U94(tt(),V1,V2)) -> mark#(U95(isNat(V1),V2))
p90: mark#(U44(X1,X2,X3)) -> mark#(X1)
p91: mark#(U45(X1,X2)) -> active#(U45(mark(X1),X2))
p92: active#(U95(tt(),V2)) -> mark#(U96(isNatList(V2)))
p93: mark#(U45(X1,X2)) -> mark#(X1)
p94: active#(isNat(length(V1))) -> mark#(U11(isNatIListKind(V1),V1))
p95: mark#(U46(X)) -> mark#(X)
p96: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p97: active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1))
p98: mark#(U51(X1,X2)) -> mark#(X1)
p99: mark#(U52(X)) -> mark#(X)
p100: mark#(U61(X1,X2)) -> active#(U61(mark(X1),X2))
p101: active#(isNatIList(V)) -> mark#(U31(isNatIListKind(V),V))
p102: mark#(U61(X1,X2)) -> mark#(X1)
p103: mark#(U62(X)) -> mark#(X)
p104: mark#(U71(X)) -> mark#(X)
p105: mark#(U81(X)) -> mark#(X)
p106: mark#(U91(X1,X2,X3)) -> active#(U91(mark(X1),X2,X3))
p107: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNatKind(V1),V1,V2))
p108: mark#(U91(X1,X2,X3)) -> mark#(X1)
p109: mark#(U92(X1,X2,X3)) -> active#(U92(mark(X1),X2,X3))
p110: active#(isNatIListKind(cons(V1,V2))) -> mark#(U51(isNatKind(V1),V2))
p111: mark#(U92(X1,X2,X3)) -> mark#(X1)
p112: mark#(U93(X1,X2,X3)) -> active#(U93(mark(X1),X2,X3))
p113: active#(isNatIListKind(take(V1,V2))) -> mark#(U61(isNatKind(V1),V2))
p114: mark#(U93(X1,X2,X3)) -> mark#(X1)
p115: mark#(U94(X1,X2,X3)) -> active#(U94(mark(X1),X2,X3))
p116: active#(isNatKind(length(V1))) -> mark#(U71(isNatIListKind(V1)))
p117: mark#(U94(X1,X2,X3)) -> mark#(X1)
p118: mark#(U95(X1,X2)) -> active#(U95(mark(X1),X2))
p119: active#(isNatKind(s(V1))) -> mark#(U81(isNatKind(V1)))
p120: mark#(U95(X1,X2)) -> mark#(X1)
p121: active#(isNatList(cons(V1,V2))) -> mark#(U91(isNatKind(V1),V1,V2))
p122: active#(isNatList(take(V1,V2))) -> mark#(U101(isNatKind(V1),V1,V2))
p123: active#(length(cons(N,L))) -> mark#(U111(isNatList(L),L,N))
p124: active#(take(|0|(),IL)) -> mark#(U121(isNatIList(IL),IL))
p125: active#(take(s(M),cons(N,IL))) -> mark#(U131(isNatIList(IL),IL,M,N))

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19, p20, p21, p22, p23, p24, p25, p26, p27, p28, p29, p30, p31, p32, p33, p34, p35, p36, p37, p38, p39, p40, p41, p42, p43, p44, p45, p46, p47, p48, p49, p50, p51, p52, p53, p54, p55, p56, p57, p58, p59, p60, p61, p62, p63, p64, p65, p66, p67, p68, p69, p70, p71, p72, p73, p74, p75, p76, p77, p78, p79, p80, p81, p82, p83, p84, p85, p86, p87, p88, p89, p90, p91, p92, p93, p94, p95, p96, p97, p98, p99, p100, p101, p102, p103, p104, p105, p106, p107, p108, p109, p110, p111, p112, p113, p114, p115, p116, p117, p118, p119, p120, p121, p122, p123, p124, p125}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(U96(X)) -> mark#(X)
p2: mark#(U95(X1,X2)) -> mark#(X1)
p3: mark#(U95(X1,X2)) -> active#(U95(mark(X1),X2))
p4: active#(take(s(M),cons(N,IL))) -> mark#(U131(isNatIList(IL),IL,M,N))
p5: mark#(U94(X1,X2,X3)) -> mark#(X1)
p6: mark#(U94(X1,X2,X3)) -> active#(U94(mark(X1),X2,X3))
p7: active#(take(|0|(),IL)) -> mark#(U121(isNatIList(IL),IL))
p8: mark#(U93(X1,X2,X3)) -> mark#(X1)
p9: mark#(U93(X1,X2,X3)) -> active#(U93(mark(X1),X2,X3))
p10: active#(length(cons(N,L))) -> mark#(U111(isNatList(L),L,N))
p11: mark#(U92(X1,X2,X3)) -> mark#(X1)
p12: mark#(U92(X1,X2,X3)) -> active#(U92(mark(X1),X2,X3))
p13: active#(isNatList(take(V1,V2))) -> mark#(U101(isNatKind(V1),V1,V2))
p14: mark#(U91(X1,X2,X3)) -> mark#(X1)
p15: mark#(U91(X1,X2,X3)) -> active#(U91(mark(X1),X2,X3))
p16: active#(isNatList(cons(V1,V2))) -> mark#(U91(isNatKind(V1),V1,V2))
p17: mark#(U81(X)) -> mark#(X)
p18: mark#(U71(X)) -> mark#(X)
p19: mark#(U62(X)) -> mark#(X)
p20: mark#(U61(X1,X2)) -> mark#(X1)
p21: mark#(U61(X1,X2)) -> active#(U61(mark(X1),X2))
p22: active#(isNatKind(s(V1))) -> mark#(U81(isNatKind(V1)))
p23: mark#(U52(X)) -> mark#(X)
p24: mark#(U51(X1,X2)) -> mark#(X1)
p25: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p26: active#(isNatKind(length(V1))) -> mark#(U71(isNatIListKind(V1)))
p27: mark#(U46(X)) -> mark#(X)
p28: mark#(U45(X1,X2)) -> mark#(X1)
p29: mark#(U45(X1,X2)) -> active#(U45(mark(X1),X2))
p30: active#(isNatIListKind(take(V1,V2))) -> mark#(U61(isNatKind(V1),V2))
p31: mark#(U44(X1,X2,X3)) -> mark#(X1)
p32: mark#(U44(X1,X2,X3)) -> active#(U44(mark(X1),X2,X3))
p33: active#(isNatIListKind(cons(V1,V2))) -> mark#(U51(isNatKind(V1),V2))
p34: mark#(U43(X1,X2,X3)) -> mark#(X1)
p35: mark#(U43(X1,X2,X3)) -> active#(U43(mark(X1),X2,X3))
p36: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNatKind(V1),V1,V2))
p37: mark#(U42(X1,X2,X3)) -> mark#(X1)
p38: mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3))
p39: active#(isNatIList(V)) -> mark#(U31(isNatIListKind(V),V))
p40: mark#(U41(X1,X2,X3)) -> mark#(X1)
p41: mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3))
p42: active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1))
p43: mark#(U33(X)) -> mark#(X)
p44: mark#(U32(X1,X2)) -> mark#(X1)
p45: mark#(U32(X1,X2)) -> active#(U32(mark(X1),X2))
p46: active#(isNat(length(V1))) -> mark#(U11(isNatIListKind(V1),V1))
p47: mark#(U31(X1,X2)) -> mark#(X1)
p48: mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2))
p49: active#(U95(tt(),V2)) -> mark#(U96(isNatList(V2)))
p50: mark#(U23(X)) -> mark#(X)
p51: mark#(U22(X1,X2)) -> mark#(X1)
p52: mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2))
p53: active#(U94(tt(),V1,V2)) -> mark#(U95(isNat(V1),V2))
p54: mark#(U21(X1,X2)) -> mark#(X1)
p55: mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2))
p56: active#(U93(tt(),V1,V2)) -> mark#(U94(isNatIListKind(V2),V1,V2))
p57: mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2)))
p58: active#(U92(tt(),V1,V2)) -> mark#(U93(isNatIListKind(V2),V1,V2))
p59: mark#(U136(X1,X2,X3,X4)) -> active#(U136(mark(X1),X2,X3,X4))
p60: active#(U91(tt(),V1,V2)) -> mark#(U92(isNatKind(V1),V1,V2))
p61: mark#(U135(X1,X2,X3,X4)) -> active#(U135(mark(X1),X2,X3,X4))
p62: active#(U61(tt(),V2)) -> mark#(U62(isNatIListKind(V2)))
p63: mark#(U134(X1,X2,X3,X4)) -> active#(U134(mark(X1),X2,X3,X4))
p64: active#(U51(tt(),V2)) -> mark#(U52(isNatIListKind(V2)))
p65: mark#(U133(X1,X2,X3,X4)) -> active#(U133(mark(X1),X2,X3,X4))
p66: active#(U45(tt(),V2)) -> mark#(U46(isNatIList(V2)))
p67: mark#(U132(X1,X2,X3,X4)) -> active#(U132(mark(X1),X2,X3,X4))
p68: active#(U44(tt(),V1,V2)) -> mark#(U45(isNat(V1),V2))
p69: mark#(U131(X1,X2,X3,X4)) -> mark#(X1)
p70: mark#(U131(X1,X2,X3,X4)) -> active#(U131(mark(X1),X2,X3,X4))
p71: active#(U43(tt(),V1,V2)) -> mark#(U44(isNatIListKind(V2),V1,V2))
p72: mark#(U122(X)) -> mark#(X)
p73: mark#(U121(X1,X2)) -> active#(U121(mark(X1),X2))
p74: active#(U42(tt(),V1,V2)) -> mark#(U43(isNatIListKind(V2),V1,V2))
p75: mark#(isNatList(X)) -> active#(isNatList(X))
p76: active#(U41(tt(),V1,V2)) -> mark#(U42(isNatKind(V1),V1,V2))
p77: mark#(U13(X)) -> mark#(X)
p78: mark#(length(X)) -> mark#(X)
p79: mark#(length(X)) -> active#(length(mark(X)))
p80: active#(U32(tt(),V)) -> mark#(U33(isNatList(V)))
p81: mark#(s(X)) -> mark#(X)
p82: mark#(U114(X1,X2)) -> active#(U114(mark(X1),X2))
p83: active#(U31(tt(),V)) -> mark#(U32(isNatIListKind(V),V))
p84: mark#(U113(X1,X2,X3)) -> active#(U113(mark(X1),X2,X3))
p85: active#(U22(tt(),V1)) -> mark#(U23(isNat(V1)))
p86: mark#(U112(X1,X2,X3)) -> active#(U112(mark(X1),X2,X3))
p87: active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1))
p88: mark#(U111(X1,X2,X3)) -> active#(U111(mark(X1),X2,X3))
p89: active#(U136(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL)))
p90: mark#(U12(X1,X2)) -> mark#(X1)
p91: mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2))
p92: active#(U135(tt(),IL,M,N)) -> mark#(U136(isNatKind(N),IL,M,N))
p93: mark#(U11(X1,X2)) -> mark#(X1)
p94: mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2))
p95: active#(U134(tt(),IL,M,N)) -> mark#(U135(isNat(N),IL,M,N))
p96: mark#(isNatIList(X)) -> active#(isNatIList(X))
p97: active#(U133(tt(),IL,M,N)) -> mark#(U134(isNatKind(M),IL,M,N))
p98: mark#(U106(X)) -> mark#(X)
p99: mark#(isNat(X)) -> active#(isNat(X))
p100: active#(U132(tt(),IL,M,N)) -> mark#(U133(isNat(M),IL,M,N))
p101: mark#(U105(X1,X2)) -> mark#(X1)
p102: mark#(U105(X1,X2)) -> active#(U105(mark(X1),X2))
p103: active#(U131(tt(),IL,M,N)) -> mark#(U132(isNatIListKind(IL),IL,M,N))
p104: mark#(U104(X1,X2,X3)) -> mark#(X1)
p105: mark#(U104(X1,X2,X3)) -> active#(U104(mark(X1),X2,X3))
p106: active#(U121(tt(),IL)) -> mark#(U122(isNatIListKind(IL)))
p107: mark#(isNatIListKind(X)) -> active#(isNatIListKind(X))
p108: active#(U12(tt(),V1)) -> mark#(U13(isNatList(V1)))
p109: mark#(U103(X1,X2,X3)) -> mark#(X1)
p110: mark#(U103(X1,X2,X3)) -> active#(U103(mark(X1),X2,X3))
p111: active#(U114(tt(),L)) -> mark#(s(length(L)))
p112: mark#(isNatKind(X)) -> active#(isNatKind(X))
p113: active#(U113(tt(),L,N)) -> mark#(U114(isNatKind(N),L))
p114: mark#(U102(X1,X2,X3)) -> mark#(X1)
p115: mark#(U102(X1,X2,X3)) -> active#(U102(mark(X1),X2,X3))
p116: active#(U112(tt(),L,N)) -> mark#(U113(isNat(N),L,N))
p117: mark#(U101(X1,X2,X3)) -> mark#(X1)
p118: mark#(U101(X1,X2,X3)) -> active#(U101(mark(X1),X2,X3))
p119: active#(U111(tt(),L,N)) -> mark#(U112(isNatIListKind(L),L,N))
p120: active#(U11(tt(),V1)) -> mark#(U12(isNatIListKind(V1),V1))
p121: active#(U105(tt(),V2)) -> mark#(U106(isNatIList(V2)))
p122: active#(U104(tt(),V1,V2)) -> mark#(U105(isNat(V1),V2))
p123: active#(U103(tt(),V1,V2)) -> mark#(U104(isNatIListKind(V2),V1,V2))
p124: active#(U102(tt(),V1,V2)) -> mark#(U103(isNatIListKind(V2),V1,V2))
p125: active#(U101(tt(),V1,V2)) -> mark#(U102(isNatKind(V1),V1,V2))

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41, r42, r43, r44, r45, r46, r47, r48, r49, r50, r51, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76, r77, r78, r79, r80, r81, r82, r83, r84, r85, r86, r87, r88, r89, r90, r91, r92, r93, r94, r95, r96, r97, r98, r99, r100, r101, r102, r103, r104, r105, r106, r107, r108, r109, r110, r111, r112, r113, r114, r115, r116, r117, r118, r119, r120, r121, r122, r123, r124, r125, r126, r127, r128, r129, r130, r131, r132, r133, r134, r135, r136, r137, r138, r139, r140, r141, r142, r143, r144, r145, r146, r147, r148, r149, r150, r151, r152, r153, r154, r155, r156, r157, r158, r159, r160, r161, r162, r163, r164, r165, r166, r167, r168, r169, r170, r171, r172, r173, r174, r175, r176, r177, r178, r179, r180, r181, r182, r183, r184, r185, r186, r187, r188, r189, r190, r191, r192, r193, r194, r195, r196, r197, r198, r199, r200, r201, r202, r203, r204, r205, r206, r207, r208, r209, r210, r211, r212, r213, r214, r215, r216, r217, r218, r219, r220, r221, r222, r223, r224, r225, r226, r227, r228, r229, r230, r231, r232, r233, r234, r235, r236, r237, r238, r239, r240, r241, r242, r243, r244, r245, r246, r247, r248, r249, r250, r251, r252, r253, r254, r255, r256, r257, r258, r259, r260, r261, r262, r263, r264, r265, r266, r267, r268, r269, r270, r271, r272, r273, r274, r275, r276, r277, r278, r279, r280, r281, r282, r283, r284, r285, r286, r287, r288, r289, r290, r291, r292, r293, r294, r295, r296, r297, r298, r299, r300, r301, r302, r303, r304, r305, r306, r307, r308, r309, r310, r311, r312, r313, r314, r315, r316, r317, r318, r319, r320, r321, r322, r323, r324, r325, r326, r327, r328, r329, r330, r331, r332, r333, r334, r335, r336, r337, r338, r339, r340, r341, r342, r343, r344, r345, r346, r347, r348, r349, r350, r351, r352, r353, r354, r355, r356, r357, r358

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    mark#_A(x1) = x1 + 1
    U96_A(x1) = max{126, x1}
    U95_A(x1,x2) = max{x1, x2 + 126}
    active#_A(x1) = max{9, x1 + 1}
    mark_A(x1) = max{5, x1}
    take_A(x1,x2) = max{166, x1 + 161, x2 + 157}
    s_A(x1) = max{146, x1}
    cons_A(x1,x2) = max{74, x1 + 69, x2}
    U131_A(x1,x2,x3,x4) = max{x1 + 10, x2 + 157, x3 + 161, x4 + 226}
    isNatIList_A(x1) = x1 + 146
    U94_A(x1,x2,x3) = max{x1 + 46, x2 + 66, x3 + 126}
    |0|_A = 149
    U121_A(x1,x2) = max{x1 + 10, x2 + 157}
    U93_A(x1,x2,x3) = max{x1 + 30, x2 + 66, x3 + 126}
    length_A(x1) = max{148, x1 + 135}
    U111_A(x1,x2,x3) = max{x1 + 9, x2 + 135, x3 + 203}
    isNatList_A(x1) = x1 + 126
    U92_A(x1,x2,x3) = max{x1, x2 + 66, x3 + 126}
    U101_A(x1,x2,x3) = max{x1 + 268, x2 + 274, x3 + 283}
    isNatKind_A(x1) = x1 + 19
    U91_A(x1,x2,x3) = max{x1, x2 + 195, x3 + 126}
    U81_A(x1) = max{6, x1}
    U71_A(x1) = max{167, x1}
    U62_A(x1) = max{237, x1}
    U61_A(x1,x2) = max{x1 + 221, x2 + 237}
    U52_A(x1) = max{15, x1}
    U51_A(x1,x2) = max{x1, x2 + 80}
    isNatIListKind_A(x1) = x1 + 80
    U46_A(x1) = max{10, x1}
    U45_A(x1,x2) = max{x1, x2 + 146}
    U44_A(x1,x2,x3) = max{x1 + 58, x2 + 67, x3 + 146}
    U43_A(x1,x2,x3) = max{x1 + 66, x2 + 72, x3 + 146}
    U41_A(x1,x2,x3) = max{x1 + 47, x2 + 74, x3 + 146}
    U42_A(x1,x2,x3) = max{x1, x2 + 73, x3 + 146}
    U31_A(x1,x2) = max{x1 + 66, x2 + 126}
    isNat_A(x1) = x1 + 66
    U21_A(x1,x2) = max{x1, x2 + 66}
    U33_A(x1) = max{15, x1}
    U32_A(x1,x2) = max{x1, x2 + 126}
    U11_A(x1,x2) = max{x1 + 9, x2 + 201}
    tt_A = 16
    U23_A(x1) = max{9, x1}
    U22_A(x1,x2) = max{x1, x2 + 66}
    U136_A(x1,x2,x3,x4) = max{x1 + 147, x2 + 157, x3 + 161, x4 + 166}
    U135_A(x1,x2,x3,x4) = max{x1 + 151, x2 + 157, x3 + 161, x4 + 166}
    U134_A(x1,x2,x3,x4) = max{x1 + 133, x2 + 157, x3 + 161, x4 + 217}
    U133_A(x1,x2,x3,x4) = max{x1 + 95, x2 + 157, x3 + 161, x4 + 217}
    U132_A(x1,x2,x3,x4) = max{x1 + 77, x2 + 157, x3 + 161, x4 + 217}
    U122_A(x1) = max{157, x1}
    U13_A(x1) = max{201, x1}
    U114_A(x1,x2) = max{151, x1 + 131, x2 + 135}
    U113_A(x1,x2,x3) = max{x1 + 135, x2 + 135, x3 + 151}
    U112_A(x1,x2,x3) = max{x1 + 8, x2 + 135, x3 + 202}
    U12_A(x1,x2) = max{x1, x2 + 201}
    U106_A(x1) = max{147, x1}
    U105_A(x1,x2) = max{x1 + 80, x2 + 147}
    U104_A(x1,x2,x3) = max{x1 + 9, x2 + 146, x3 + 148}
    U103_A(x1,x2,x3) = max{x1, x2 + 146, x3 + 149}
    U102_A(x1,x2,x3) = max{x1, x2 + 274, x3 + 149}
    active_A(x1) = max{6, x1}
    zeros_A = 218
    nil_A = 17

The next rules are strictly ordered:

  p5, p8, p20, p31, p34, p40, p47, p69, p78, p93, p101, p104, p117

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(U96(X)) -> mark#(X)
p2: mark#(U95(X1,X2)) -> mark#(X1)
p3: mark#(U95(X1,X2)) -> active#(U95(mark(X1),X2))
p4: active#(take(s(M),cons(N,IL))) -> mark#(U131(isNatIList(IL),IL,M,N))
p5: mark#(U94(X1,X2,X3)) -> active#(U94(mark(X1),X2,X3))
p6: active#(take(|0|(),IL)) -> mark#(U121(isNatIList(IL),IL))
p7: mark#(U93(X1,X2,X3)) -> active#(U93(mark(X1),X2,X3))
p8: active#(length(cons(N,L))) -> mark#(U111(isNatList(L),L,N))
p9: mark#(U92(X1,X2,X3)) -> mark#(X1)
p10: mark#(U92(X1,X2,X3)) -> active#(U92(mark(X1),X2,X3))
p11: active#(isNatList(take(V1,V2))) -> mark#(U101(isNatKind(V1),V1,V2))
p12: mark#(U91(X1,X2,X3)) -> mark#(X1)
p13: mark#(U91(X1,X2,X3)) -> active#(U91(mark(X1),X2,X3))
p14: active#(isNatList(cons(V1,V2))) -> mark#(U91(isNatKind(V1),V1,V2))
p15: mark#(U81(X)) -> mark#(X)
p16: mark#(U71(X)) -> mark#(X)
p17: mark#(U62(X)) -> mark#(X)
p18: mark#(U61(X1,X2)) -> active#(U61(mark(X1),X2))
p19: active#(isNatKind(s(V1))) -> mark#(U81(isNatKind(V1)))
p20: mark#(U52(X)) -> mark#(X)
p21: mark#(U51(X1,X2)) -> mark#(X1)
p22: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p23: active#(isNatKind(length(V1))) -> mark#(U71(isNatIListKind(V1)))
p24: mark#(U46(X)) -> mark#(X)
p25: mark#(U45(X1,X2)) -> mark#(X1)
p26: mark#(U45(X1,X2)) -> active#(U45(mark(X1),X2))
p27: active#(isNatIListKind(take(V1,V2))) -> mark#(U61(isNatKind(V1),V2))
p28: mark#(U44(X1,X2,X3)) -> active#(U44(mark(X1),X2,X3))
p29: active#(isNatIListKind(cons(V1,V2))) -> mark#(U51(isNatKind(V1),V2))
p30: mark#(U43(X1,X2,X3)) -> active#(U43(mark(X1),X2,X3))
p31: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNatKind(V1),V1,V2))
p32: mark#(U42(X1,X2,X3)) -> mark#(X1)
p33: mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3))
p34: active#(isNatIList(V)) -> mark#(U31(isNatIListKind(V),V))
p35: mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3))
p36: active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1))
p37: mark#(U33(X)) -> mark#(X)
p38: mark#(U32(X1,X2)) -> mark#(X1)
p39: mark#(U32(X1,X2)) -> active#(U32(mark(X1),X2))
p40: active#(isNat(length(V1))) -> mark#(U11(isNatIListKind(V1),V1))
p41: mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2))
p42: active#(U95(tt(),V2)) -> mark#(U96(isNatList(V2)))
p43: mark#(U23(X)) -> mark#(X)
p44: mark#(U22(X1,X2)) -> mark#(X1)
p45: mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2))
p46: active#(U94(tt(),V1,V2)) -> mark#(U95(isNat(V1),V2))
p47: mark#(U21(X1,X2)) -> mark#(X1)
p48: mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2))
p49: active#(U93(tt(),V1,V2)) -> mark#(U94(isNatIListKind(V2),V1,V2))
p50: mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2)))
p51: active#(U92(tt(),V1,V2)) -> mark#(U93(isNatIListKind(V2),V1,V2))
p52: mark#(U136(X1,X2,X3,X4)) -> active#(U136(mark(X1),X2,X3,X4))
p53: active#(U91(tt(),V1,V2)) -> mark#(U92(isNatKind(V1),V1,V2))
p54: mark#(U135(X1,X2,X3,X4)) -> active#(U135(mark(X1),X2,X3,X4))
p55: active#(U61(tt(),V2)) -> mark#(U62(isNatIListKind(V2)))
p56: mark#(U134(X1,X2,X3,X4)) -> active#(U134(mark(X1),X2,X3,X4))
p57: active#(U51(tt(),V2)) -> mark#(U52(isNatIListKind(V2)))
p58: mark#(U133(X1,X2,X3,X4)) -> active#(U133(mark(X1),X2,X3,X4))
p59: active#(U45(tt(),V2)) -> mark#(U46(isNatIList(V2)))
p60: mark#(U132(X1,X2,X3,X4)) -> active#(U132(mark(X1),X2,X3,X4))
p61: active#(U44(tt(),V1,V2)) -> mark#(U45(isNat(V1),V2))
p62: mark#(U131(X1,X2,X3,X4)) -> active#(U131(mark(X1),X2,X3,X4))
p63: active#(U43(tt(),V1,V2)) -> mark#(U44(isNatIListKind(V2),V1,V2))
p64: mark#(U122(X)) -> mark#(X)
p65: mark#(U121(X1,X2)) -> active#(U121(mark(X1),X2))
p66: active#(U42(tt(),V1,V2)) -> mark#(U43(isNatIListKind(V2),V1,V2))
p67: mark#(isNatList(X)) -> active#(isNatList(X))
p68: active#(U41(tt(),V1,V2)) -> mark#(U42(isNatKind(V1),V1,V2))
p69: mark#(U13(X)) -> mark#(X)
p70: mark#(length(X)) -> active#(length(mark(X)))
p71: active#(U32(tt(),V)) -> mark#(U33(isNatList(V)))
p72: mark#(s(X)) -> mark#(X)
p73: mark#(U114(X1,X2)) -> active#(U114(mark(X1),X2))
p74: active#(U31(tt(),V)) -> mark#(U32(isNatIListKind(V),V))
p75: mark#(U113(X1,X2,X3)) -> active#(U113(mark(X1),X2,X3))
p76: active#(U22(tt(),V1)) -> mark#(U23(isNat(V1)))
p77: mark#(U112(X1,X2,X3)) -> active#(U112(mark(X1),X2,X3))
p78: active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1))
p79: mark#(U111(X1,X2,X3)) -> active#(U111(mark(X1),X2,X3))
p80: active#(U136(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL)))
p81: mark#(U12(X1,X2)) -> mark#(X1)
p82: mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2))
p83: active#(U135(tt(),IL,M,N)) -> mark#(U136(isNatKind(N),IL,M,N))
p84: mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2))
p85: active#(U134(tt(),IL,M,N)) -> mark#(U135(isNat(N),IL,M,N))
p86: mark#(isNatIList(X)) -> active#(isNatIList(X))
p87: active#(U133(tt(),IL,M,N)) -> mark#(U134(isNatKind(M),IL,M,N))
p88: mark#(U106(X)) -> mark#(X)
p89: mark#(isNat(X)) -> active#(isNat(X))
p90: active#(U132(tt(),IL,M,N)) -> mark#(U133(isNat(M),IL,M,N))
p91: mark#(U105(X1,X2)) -> active#(U105(mark(X1),X2))
p92: active#(U131(tt(),IL,M,N)) -> mark#(U132(isNatIListKind(IL),IL,M,N))
p93: mark#(U104(X1,X2,X3)) -> active#(U104(mark(X1),X2,X3))
p94: active#(U121(tt(),IL)) -> mark#(U122(isNatIListKind(IL)))
p95: mark#(isNatIListKind(X)) -> active#(isNatIListKind(X))
p96: active#(U12(tt(),V1)) -> mark#(U13(isNatList(V1)))
p97: mark#(U103(X1,X2,X3)) -> mark#(X1)
p98: mark#(U103(X1,X2,X3)) -> active#(U103(mark(X1),X2,X3))
p99: active#(U114(tt(),L)) -> mark#(s(length(L)))
p100: mark#(isNatKind(X)) -> active#(isNatKind(X))
p101: active#(U113(tt(),L,N)) -> mark#(U114(isNatKind(N),L))
p102: mark#(U102(X1,X2,X3)) -> mark#(X1)
p103: mark#(U102(X1,X2,X3)) -> active#(U102(mark(X1),X2,X3))
p104: active#(U112(tt(),L,N)) -> mark#(U113(isNat(N),L,N))
p105: mark#(U101(X1,X2,X3)) -> active#(U101(mark(X1),X2,X3))
p106: active#(U111(tt(),L,N)) -> mark#(U112(isNatIListKind(L),L,N))
p107: active#(U11(tt(),V1)) -> mark#(U12(isNatIListKind(V1),V1))
p108: active#(U105(tt(),V2)) -> mark#(U106(isNatIList(V2)))
p109: active#(U104(tt(),V1,V2)) -> mark#(U105(isNat(V1),V2))
p110: active#(U103(tt(),V1,V2)) -> mark#(U104(isNatIListKind(V2),V1,V2))
p111: active#(U102(tt(),V1,V2)) -> mark#(U103(isNatIListKind(V2),V1,V2))
p112: active#(U101(tt(),V1,V2)) -> mark#(U102(isNatKind(V1),V1,V2))

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19, p20, p21, p22, p23, p24, p25, p26, p27, p28, p29, p30, p31, p32, p33, p34, p35, p36, p37, p38, p39, p40, p41, p42, p43, p44, p45, p46, p47, p48, p49, p50, p51, p52, p53, p54, p55, p56, p57, p58, p59, p60, p61, p62, p63, p64, p65, p66, p67, p68, p69, p70, p71, p72, p73, p74, p75, p76, p77, p78, p79, p80, p81, p82, p83, p84, p85, p86, p87, p88, p89, p90, p91, p92, p93, p94, p95, p96, p97, p98, p99, p100, p101, p102, p103, p104, p105, p106, p107, p108, p109, p110, p111, p112}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(U96(X)) -> mark#(X)
p2: mark#(U101(X1,X2,X3)) -> active#(U101(mark(X1),X2,X3))
p3: active#(U101(tt(),V1,V2)) -> mark#(U102(isNatKind(V1),V1,V2))
p4: mark#(U102(X1,X2,X3)) -> active#(U102(mark(X1),X2,X3))
p5: active#(U102(tt(),V1,V2)) -> mark#(U103(isNatIListKind(V2),V1,V2))
p6: mark#(U102(X1,X2,X3)) -> mark#(X1)
p7: mark#(isNatKind(X)) -> active#(isNatKind(X))
p8: active#(U103(tt(),V1,V2)) -> mark#(U104(isNatIListKind(V2),V1,V2))
p9: mark#(U103(X1,X2,X3)) -> active#(U103(mark(X1),X2,X3))
p10: active#(U104(tt(),V1,V2)) -> mark#(U105(isNat(V1),V2))
p11: mark#(U103(X1,X2,X3)) -> mark#(X1)
p12: mark#(isNatIListKind(X)) -> active#(isNatIListKind(X))
p13: active#(U105(tt(),V2)) -> mark#(U106(isNatIList(V2)))
p14: mark#(U104(X1,X2,X3)) -> active#(U104(mark(X1),X2,X3))
p15: active#(U11(tt(),V1)) -> mark#(U12(isNatIListKind(V1),V1))
p16: mark#(U105(X1,X2)) -> active#(U105(mark(X1),X2))
p17: active#(U111(tt(),L,N)) -> mark#(U112(isNatIListKind(L),L,N))
p18: mark#(isNat(X)) -> active#(isNat(X))
p19: active#(U112(tt(),L,N)) -> mark#(U113(isNat(N),L,N))
p20: mark#(U106(X)) -> mark#(X)
p21: mark#(isNatIList(X)) -> active#(isNatIList(X))
p22: active#(U113(tt(),L,N)) -> mark#(U114(isNatKind(N),L))
p23: mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2))
p24: active#(U114(tt(),L)) -> mark#(s(length(L)))
p25: mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2))
p26: active#(U12(tt(),V1)) -> mark#(U13(isNatList(V1)))
p27: mark#(U12(X1,X2)) -> mark#(X1)
p28: mark#(U111(X1,X2,X3)) -> active#(U111(mark(X1),X2,X3))
p29: active#(U121(tt(),IL)) -> mark#(U122(isNatIListKind(IL)))
p30: mark#(U112(X1,X2,X3)) -> active#(U112(mark(X1),X2,X3))
p31: active#(U131(tt(),IL,M,N)) -> mark#(U132(isNatIListKind(IL),IL,M,N))
p32: mark#(U113(X1,X2,X3)) -> active#(U113(mark(X1),X2,X3))
p33: active#(U132(tt(),IL,M,N)) -> mark#(U133(isNat(M),IL,M,N))
p34: mark#(U114(X1,X2)) -> active#(U114(mark(X1),X2))
p35: active#(U133(tt(),IL,M,N)) -> mark#(U134(isNatKind(M),IL,M,N))
p36: mark#(s(X)) -> mark#(X)
p37: mark#(length(X)) -> active#(length(mark(X)))
p38: active#(U134(tt(),IL,M,N)) -> mark#(U135(isNat(N),IL,M,N))
p39: mark#(U13(X)) -> mark#(X)
p40: mark#(isNatList(X)) -> active#(isNatList(X))
p41: active#(U135(tt(),IL,M,N)) -> mark#(U136(isNatKind(N),IL,M,N))
p42: mark#(U121(X1,X2)) -> active#(U121(mark(X1),X2))
p43: active#(U136(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL)))
p44: mark#(U122(X)) -> mark#(X)
p45: mark#(U131(X1,X2,X3,X4)) -> active#(U131(mark(X1),X2,X3,X4))
p46: active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1))
p47: mark#(U132(X1,X2,X3,X4)) -> active#(U132(mark(X1),X2,X3,X4))
p48: active#(U22(tt(),V1)) -> mark#(U23(isNat(V1)))
p49: mark#(U133(X1,X2,X3,X4)) -> active#(U133(mark(X1),X2,X3,X4))
p50: active#(U31(tt(),V)) -> mark#(U32(isNatIListKind(V),V))
p51: mark#(U134(X1,X2,X3,X4)) -> active#(U134(mark(X1),X2,X3,X4))
p52: active#(U32(tt(),V)) -> mark#(U33(isNatList(V)))
p53: mark#(U135(X1,X2,X3,X4)) -> active#(U135(mark(X1),X2,X3,X4))
p54: active#(U41(tt(),V1,V2)) -> mark#(U42(isNatKind(V1),V1,V2))
p55: mark#(U136(X1,X2,X3,X4)) -> active#(U136(mark(X1),X2,X3,X4))
p56: active#(U42(tt(),V1,V2)) -> mark#(U43(isNatIListKind(V2),V1,V2))
p57: mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2)))
p58: active#(U43(tt(),V1,V2)) -> mark#(U44(isNatIListKind(V2),V1,V2))
p59: mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2))
p60: active#(U44(tt(),V1,V2)) -> mark#(U45(isNat(V1),V2))
p61: mark#(U21(X1,X2)) -> mark#(X1)
p62: mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2))
p63: active#(U45(tt(),V2)) -> mark#(U46(isNatIList(V2)))
p64: mark#(U22(X1,X2)) -> mark#(X1)
p65: mark#(U23(X)) -> mark#(X)
p66: mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2))
p67: active#(U51(tt(),V2)) -> mark#(U52(isNatIListKind(V2)))
p68: mark#(U32(X1,X2)) -> active#(U32(mark(X1),X2))
p69: active#(U61(tt(),V2)) -> mark#(U62(isNatIListKind(V2)))
p70: mark#(U32(X1,X2)) -> mark#(X1)
p71: mark#(U33(X)) -> mark#(X)
p72: mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3))
p73: active#(U91(tt(),V1,V2)) -> mark#(U92(isNatKind(V1),V1,V2))
p74: mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3))
p75: active#(U92(tt(),V1,V2)) -> mark#(U93(isNatIListKind(V2),V1,V2))
p76: mark#(U42(X1,X2,X3)) -> mark#(X1)
p77: mark#(U43(X1,X2,X3)) -> active#(U43(mark(X1),X2,X3))
p78: active#(U93(tt(),V1,V2)) -> mark#(U94(isNatIListKind(V2),V1,V2))
p79: mark#(U44(X1,X2,X3)) -> active#(U44(mark(X1),X2,X3))
p80: active#(U94(tt(),V1,V2)) -> mark#(U95(isNat(V1),V2))
p81: mark#(U45(X1,X2)) -> active#(U45(mark(X1),X2))
p82: active#(U95(tt(),V2)) -> mark#(U96(isNatList(V2)))
p83: mark#(U45(X1,X2)) -> mark#(X1)
p84: mark#(U46(X)) -> mark#(X)
p85: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p86: active#(isNat(length(V1))) -> mark#(U11(isNatIListKind(V1),V1))
p87: mark#(U51(X1,X2)) -> mark#(X1)
p88: mark#(U52(X)) -> mark#(X)
p89: mark#(U61(X1,X2)) -> active#(U61(mark(X1),X2))
p90: active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1))
p91: mark#(U62(X)) -> mark#(X)
p92: mark#(U71(X)) -> mark#(X)
p93: mark#(U81(X)) -> mark#(X)
p94: mark#(U91(X1,X2,X3)) -> active#(U91(mark(X1),X2,X3))
p95: active#(isNatIList(V)) -> mark#(U31(isNatIListKind(V),V))
p96: mark#(U91(X1,X2,X3)) -> mark#(X1)
p97: mark#(U92(X1,X2,X3)) -> active#(U92(mark(X1),X2,X3))
p98: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNatKind(V1),V1,V2))
p99: mark#(U92(X1,X2,X3)) -> mark#(X1)
p100: mark#(U93(X1,X2,X3)) -> active#(U93(mark(X1),X2,X3))
p101: active#(isNatIListKind(cons(V1,V2))) -> mark#(U51(isNatKind(V1),V2))
p102: mark#(U94(X1,X2,X3)) -> active#(U94(mark(X1),X2,X3))
p103: active#(isNatIListKind(take(V1,V2))) -> mark#(U61(isNatKind(V1),V2))
p104: mark#(U95(X1,X2)) -> active#(U95(mark(X1),X2))
p105: active#(isNatKind(length(V1))) -> mark#(U71(isNatIListKind(V1)))
p106: mark#(U95(X1,X2)) -> mark#(X1)
p107: active#(isNatKind(s(V1))) -> mark#(U81(isNatKind(V1)))
p108: active#(isNatList(cons(V1,V2))) -> mark#(U91(isNatKind(V1),V1,V2))
p109: active#(isNatList(take(V1,V2))) -> mark#(U101(isNatKind(V1),V1,V2))
p110: active#(length(cons(N,L))) -> mark#(U111(isNatList(L),L,N))
p111: active#(take(|0|(),IL)) -> mark#(U121(isNatIList(IL),IL))
p112: active#(take(s(M),cons(N,IL))) -> mark#(U131(isNatIList(IL),IL,M,N))

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41, r42, r43, r44, r45, r46, r47, r48, r49, r50, r51, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76, r77, r78, r79, r80, r81, r82, r83, r84, r85, r86, r87, r88, r89, r90, r91, r92, r93, r94, r95, r96, r97, r98, r99, r100, r101, r102, r103, r104, r105, r106, r107, r108, r109, r110, r111, r112, r113, r114, r115, r116, r117, r118, r119, r120, r121, r122, r123, r124, r125, r126, r127, r128, r129, r130, r131, r132, r133, r134, r135, r136, r137, r138, r139, r140, r141, r142, r143, r144, r145, r146, r147, r148, r149, r150, r151, r152, r153, r154, r155, r156, r157, r158, r159, r160, r161, r162, r163, r164, r165, r166, r167, r168, r169, r170, r171, r172, r173, r174, r175, r176, r177, r178, r179, r180, r181, r182, r183, r184, r185, r186, r187, r188, r189, r190, r191, r192, r193, r194, r195, r196, r197, r198, r199, r200, r201, r202, r203, r204, r205, r206, r207, r208, r209, r210, r211, r212, r213, r214, r215, r216, r217, r218, r219, r220, r221, r222, r223, r224, r225, r226, r227, r228, r229, r230, r231, r232, r233, r234, r235, r236, r237, r238, r239, r240, r241, r242, r243, r244, r245, r246, r247, r248, r249, r250, r251, r252, r253, r254, r255, r256, r257, r258, r259, r260, r261, r262, r263, r264, r265, r266, r267, r268, r269, r270, r271, r272, r273, r274, r275, r276, r277, r278, r279, r280, r281, r282, r283, r284, r285, r286, r287, r288, r289, r290, r291, r292, r293, r294, r295, r296, r297, r298, r299, r300, r301, r302, r303, r304, r305, r306, r307, r308, r309, r310, r311, r312, r313, r314, r315, r316, r317, r318, r319, r320, r321, r322, r323, r324, r325, r326, r327, r328, r329, r330, r331, r332, r333, r334, r335, r336, r337, r338, r339, r340, r341, r342, r343, r344, r345, r346, r347, r348, r349, r350, r351, r352, r353, r354, r355, r356, r357, r358

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    mark#_A(x1) = x1 + 71
    U96_A(x1) = max{122, x1}
    U101_A(x1,x2,x3) = max{x1 + 433, x2 + 433, x3 + 433}
    active#_A(x1) = x1 + 71
    mark_A(x1) = x1
    tt_A = 122
    U102_A(x1,x2,x3) = max{x1, x2 + 433, x3 + 433}
    isNatKind_A(x1) = max{12, x1 - 69}
    U103_A(x1,x2,x3) = max{x1, x2 + 249, x3 + 363}
    isNatIListKind_A(x1) = x1 + 40
    U104_A(x1,x2,x3) = max{x1 + 124, x2 + 249, x3 + 362}
    U105_A(x1,x2) = max{x1 + 236, x2 + 361}
    isNat_A(x1) = x1 + 13
    U106_A(x1) = max{361, x1}
    isNatIList_A(x1) = x1 + 360
    U11_A(x1,x2) = max{x1 + 5, x2 + 122}
    U12_A(x1,x2) = max{x1, x2 + 122}
    U111_A(x1,x2,x3) = max{x1 + 5, x2 + 109, x3 + 111}
    U112_A(x1,x2,x3) = max{x1 + 69, x2 + 109, x3 + 110}
    U113_A(x1,x2,x3) = max{x1 + 97, x2 + 109, x3 + 110}
    U114_A(x1,x2) = max{x1 + 83, x2 + 109}
    s_A(x1) = max{81, x1}
    length_A(x1) = x1 + 109
    U13_A(x1) = max{122, x1}
    isNatList_A(x1) = x1 + 103
    U121_A(x1,x2) = max{x1 + 6, x2 + 40}
    U122_A(x1) = max{4, x1}
    U131_A(x1,x2,x3,x4) = max{x1 + 6, x2 + 366, x3 + 330, x4 + 414}
    U132_A(x1,x2,x3,x4) = max{x1 + 325, x2 + 366, x3 + 330, x4 + 413}
    U133_A(x1,x2,x3,x4) = max{x1 + 275, x2 + 366, x3 + 330, x4 + 412}
    U134_A(x1,x2,x3,x4) = max{x1 + 399, x2 + 366, x3 + 330, x4 + 397}
    U135_A(x1,x2,x3,x4) = max{x1 + 383, x2 + 366, x3 + 330, x4 + 383}
    U136_A(x1,x2,x3,x4) = max{x1 + 369, x2 + 366, x3 + 330, x4 + 383}
    cons_A(x1,x2) = max{x1 + 49, x2}
    take_A(x1,x2) = max{367, x1 + 330, x2 + 366}
    U21_A(x1,x2) = max{x1, x2 + 13}
    U22_A(x1,x2) = max{x1, x2 + 13}
    U23_A(x1) = max{4, x1}
    U31_A(x1,x2) = max{x1 + 2, x2 + 123}
    U32_A(x1,x2) = max{x1, x2 + 122}
    U33_A(x1) = max{122, x1}
    U41_A(x1,x2,x3) = max{x1 + 360, x2 + 359, x3 + 360}
    U42_A(x1,x2,x3) = max{x1, x2 + 326, x3 + 360}
    U43_A(x1,x2,x3) = max{x1 + 124, x2 + 325, x3 + 360}
    U44_A(x1,x2,x3) = max{x1 + 320, x2 + 248, x3 + 360}
    U45_A(x1,x2) = max{x1, x2 + 360}
    U46_A(x1) = max{360, x1}
    U51_A(x1,x2) = max{x1, x2 + 40}
    U52_A(x1) = max{39, x1}
    U61_A(x1,x2) = max{x1 + 41, x2 + 40}
    U62_A(x1) = max{4, x1}
    U91_A(x1,x2,x3) = max{x1, x2 + 70, x3 + 103}
    U92_A(x1,x2,x3) = max{x1, x2 + 69, x3 + 103}
    U93_A(x1,x2,x3) = max{x1 + 62, x2 + 63, x3 + 103}
    U94_A(x1,x2,x3) = max{x1 + 13, x2 + 14, x3 + 103}
    U95_A(x1,x2) = max{x1, x2 + 103}
    U71_A(x1) = max{11, x1}
    U81_A(x1) = max{11, x1}
    |0|_A = 191
    active_A(x1) = max{3, x1}
    zeros_A = 240
    nil_A = 82

The next rules are strictly ordered:

  p5, p50, p95, p103

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(U96(X)) -> mark#(X)
p2: mark#(U101(X1,X2,X3)) -> active#(U101(mark(X1),X2,X3))
p3: active#(U101(tt(),V1,V2)) -> mark#(U102(isNatKind(V1),V1,V2))
p4: mark#(U102(X1,X2,X3)) -> active#(U102(mark(X1),X2,X3))
p5: mark#(U102(X1,X2,X3)) -> mark#(X1)
p6: mark#(isNatKind(X)) -> active#(isNatKind(X))
p7: active#(U103(tt(),V1,V2)) -> mark#(U104(isNatIListKind(V2),V1,V2))
p8: mark#(U103(X1,X2,X3)) -> active#(U103(mark(X1),X2,X3))
p9: active#(U104(tt(),V1,V2)) -> mark#(U105(isNat(V1),V2))
p10: mark#(U103(X1,X2,X3)) -> mark#(X1)
p11: mark#(isNatIListKind(X)) -> active#(isNatIListKind(X))
p12: active#(U105(tt(),V2)) -> mark#(U106(isNatIList(V2)))
p13: mark#(U104(X1,X2,X3)) -> active#(U104(mark(X1),X2,X3))
p14: active#(U11(tt(),V1)) -> mark#(U12(isNatIListKind(V1),V1))
p15: mark#(U105(X1,X2)) -> active#(U105(mark(X1),X2))
p16: active#(U111(tt(),L,N)) -> mark#(U112(isNatIListKind(L),L,N))
p17: mark#(isNat(X)) -> active#(isNat(X))
p18: active#(U112(tt(),L,N)) -> mark#(U113(isNat(N),L,N))
p19: mark#(U106(X)) -> mark#(X)
p20: mark#(isNatIList(X)) -> active#(isNatIList(X))
p21: active#(U113(tt(),L,N)) -> mark#(U114(isNatKind(N),L))
p22: mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2))
p23: active#(U114(tt(),L)) -> mark#(s(length(L)))
p24: mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2))
p25: active#(U12(tt(),V1)) -> mark#(U13(isNatList(V1)))
p26: mark#(U12(X1,X2)) -> mark#(X1)
p27: mark#(U111(X1,X2,X3)) -> active#(U111(mark(X1),X2,X3))
p28: active#(U121(tt(),IL)) -> mark#(U122(isNatIListKind(IL)))
p29: mark#(U112(X1,X2,X3)) -> active#(U112(mark(X1),X2,X3))
p30: active#(U131(tt(),IL,M,N)) -> mark#(U132(isNatIListKind(IL),IL,M,N))
p31: mark#(U113(X1,X2,X3)) -> active#(U113(mark(X1),X2,X3))
p32: active#(U132(tt(),IL,M,N)) -> mark#(U133(isNat(M),IL,M,N))
p33: mark#(U114(X1,X2)) -> active#(U114(mark(X1),X2))
p34: active#(U133(tt(),IL,M,N)) -> mark#(U134(isNatKind(M),IL,M,N))
p35: mark#(s(X)) -> mark#(X)
p36: mark#(length(X)) -> active#(length(mark(X)))
p37: active#(U134(tt(),IL,M,N)) -> mark#(U135(isNat(N),IL,M,N))
p38: mark#(U13(X)) -> mark#(X)
p39: mark#(isNatList(X)) -> active#(isNatList(X))
p40: active#(U135(tt(),IL,M,N)) -> mark#(U136(isNatKind(N),IL,M,N))
p41: mark#(U121(X1,X2)) -> active#(U121(mark(X1),X2))
p42: active#(U136(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL)))
p43: mark#(U122(X)) -> mark#(X)
p44: mark#(U131(X1,X2,X3,X4)) -> active#(U131(mark(X1),X2,X3,X4))
p45: active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1))
p46: mark#(U132(X1,X2,X3,X4)) -> active#(U132(mark(X1),X2,X3,X4))
p47: active#(U22(tt(),V1)) -> mark#(U23(isNat(V1)))
p48: mark#(U133(X1,X2,X3,X4)) -> active#(U133(mark(X1),X2,X3,X4))
p49: mark#(U134(X1,X2,X3,X4)) -> active#(U134(mark(X1),X2,X3,X4))
p50: active#(U32(tt(),V)) -> mark#(U33(isNatList(V)))
p51: mark#(U135(X1,X2,X3,X4)) -> active#(U135(mark(X1),X2,X3,X4))
p52: active#(U41(tt(),V1,V2)) -> mark#(U42(isNatKind(V1),V1,V2))
p53: mark#(U136(X1,X2,X3,X4)) -> active#(U136(mark(X1),X2,X3,X4))
p54: active#(U42(tt(),V1,V2)) -> mark#(U43(isNatIListKind(V2),V1,V2))
p55: mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2)))
p56: active#(U43(tt(),V1,V2)) -> mark#(U44(isNatIListKind(V2),V1,V2))
p57: mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2))
p58: active#(U44(tt(),V1,V2)) -> mark#(U45(isNat(V1),V2))
p59: mark#(U21(X1,X2)) -> mark#(X1)
p60: mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2))
p61: active#(U45(tt(),V2)) -> mark#(U46(isNatIList(V2)))
p62: mark#(U22(X1,X2)) -> mark#(X1)
p63: mark#(U23(X)) -> mark#(X)
p64: mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2))
p65: active#(U51(tt(),V2)) -> mark#(U52(isNatIListKind(V2)))
p66: mark#(U32(X1,X2)) -> active#(U32(mark(X1),X2))
p67: active#(U61(tt(),V2)) -> mark#(U62(isNatIListKind(V2)))
p68: mark#(U32(X1,X2)) -> mark#(X1)
p69: mark#(U33(X)) -> mark#(X)
p70: mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3))
p71: active#(U91(tt(),V1,V2)) -> mark#(U92(isNatKind(V1),V1,V2))
p72: mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3))
p73: active#(U92(tt(),V1,V2)) -> mark#(U93(isNatIListKind(V2),V1,V2))
p74: mark#(U42(X1,X2,X3)) -> mark#(X1)
p75: mark#(U43(X1,X2,X3)) -> active#(U43(mark(X1),X2,X3))
p76: active#(U93(tt(),V1,V2)) -> mark#(U94(isNatIListKind(V2),V1,V2))
p77: mark#(U44(X1,X2,X3)) -> active#(U44(mark(X1),X2,X3))
p78: active#(U94(tt(),V1,V2)) -> mark#(U95(isNat(V1),V2))
p79: mark#(U45(X1,X2)) -> active#(U45(mark(X1),X2))
p80: active#(U95(tt(),V2)) -> mark#(U96(isNatList(V2)))
p81: mark#(U45(X1,X2)) -> mark#(X1)
p82: mark#(U46(X)) -> mark#(X)
p83: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p84: active#(isNat(length(V1))) -> mark#(U11(isNatIListKind(V1),V1))
p85: mark#(U51(X1,X2)) -> mark#(X1)
p86: mark#(U52(X)) -> mark#(X)
p87: mark#(U61(X1,X2)) -> active#(U61(mark(X1),X2))
p88: active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1))
p89: mark#(U62(X)) -> mark#(X)
p90: mark#(U71(X)) -> mark#(X)
p91: mark#(U81(X)) -> mark#(X)
p92: mark#(U91(X1,X2,X3)) -> active#(U91(mark(X1),X2,X3))
p93: mark#(U91(X1,X2,X3)) -> mark#(X1)
p94: mark#(U92(X1,X2,X3)) -> active#(U92(mark(X1),X2,X3))
p95: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNatKind(V1),V1,V2))
p96: mark#(U92(X1,X2,X3)) -> mark#(X1)
p97: mark#(U93(X1,X2,X3)) -> active#(U93(mark(X1),X2,X3))
p98: active#(isNatIListKind(cons(V1,V2))) -> mark#(U51(isNatKind(V1),V2))
p99: mark#(U94(X1,X2,X3)) -> active#(U94(mark(X1),X2,X3))
p100: mark#(U95(X1,X2)) -> active#(U95(mark(X1),X2))
p101: active#(isNatKind(length(V1))) -> mark#(U71(isNatIListKind(V1)))
p102: mark#(U95(X1,X2)) -> mark#(X1)
p103: active#(isNatKind(s(V1))) -> mark#(U81(isNatKind(V1)))
p104: active#(isNatList(cons(V1,V2))) -> mark#(U91(isNatKind(V1),V1,V2))
p105: active#(isNatList(take(V1,V2))) -> mark#(U101(isNatKind(V1),V1,V2))
p106: active#(length(cons(N,L))) -> mark#(U111(isNatList(L),L,N))
p107: active#(take(|0|(),IL)) -> mark#(U121(isNatIList(IL),IL))
p108: active#(take(s(M),cons(N,IL))) -> mark#(U131(isNatIList(IL),IL,M,N))

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19, p20, p21, p22, p23, p24, p25, p26, p27, p28, p29, p30, p31, p32, p33, p34, p35, p36, p37, p38, p39, p40, p41, p42, p43, p44, p45, p46, p47, p48, p49, p50, p51, p52, p53, p54, p55, p56, p57, p58, p59, p60, p61, p62, p63, p64, p65, p66, p67, p68, p69, p70, p71, p72, p73, p74, p75, p76, p77, p78, p79, p80, p81, p82, p83, p84, p85, p86, p87, p88, p89, p90, p91, p92, p93, p94, p95, p96, p97, p98, p99, p100, p101, p102, p103, p104, p105, p106, p107, p108}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(U96(X)) -> mark#(X)
p2: mark#(U95(X1,X2)) -> mark#(X1)
p3: mark#(U95(X1,X2)) -> active#(U95(mark(X1),X2))
p4: active#(take(s(M),cons(N,IL))) -> mark#(U131(isNatIList(IL),IL,M,N))
p5: mark#(U94(X1,X2,X3)) -> active#(U94(mark(X1),X2,X3))
p6: active#(take(|0|(),IL)) -> mark#(U121(isNatIList(IL),IL))
p7: mark#(U93(X1,X2,X3)) -> active#(U93(mark(X1),X2,X3))
p8: active#(length(cons(N,L))) -> mark#(U111(isNatList(L),L,N))
p9: mark#(U92(X1,X2,X3)) -> mark#(X1)
p10: mark#(U92(X1,X2,X3)) -> active#(U92(mark(X1),X2,X3))
p11: active#(isNatList(take(V1,V2))) -> mark#(U101(isNatKind(V1),V1,V2))
p12: mark#(U91(X1,X2,X3)) -> mark#(X1)
p13: mark#(U91(X1,X2,X3)) -> active#(U91(mark(X1),X2,X3))
p14: active#(isNatList(cons(V1,V2))) -> mark#(U91(isNatKind(V1),V1,V2))
p15: mark#(U81(X)) -> mark#(X)
p16: mark#(U71(X)) -> mark#(X)
p17: mark#(U62(X)) -> mark#(X)
p18: mark#(U61(X1,X2)) -> active#(U61(mark(X1),X2))
p19: active#(isNatKind(s(V1))) -> mark#(U81(isNatKind(V1)))
p20: mark#(U52(X)) -> mark#(X)
p21: mark#(U51(X1,X2)) -> mark#(X1)
p22: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p23: active#(isNatKind(length(V1))) -> mark#(U71(isNatIListKind(V1)))
p24: mark#(U46(X)) -> mark#(X)
p25: mark#(U45(X1,X2)) -> mark#(X1)
p26: mark#(U45(X1,X2)) -> active#(U45(mark(X1),X2))
p27: active#(isNatIListKind(cons(V1,V2))) -> mark#(U51(isNatKind(V1),V2))
p28: mark#(U44(X1,X2,X3)) -> active#(U44(mark(X1),X2,X3))
p29: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNatKind(V1),V1,V2))
p30: mark#(U43(X1,X2,X3)) -> active#(U43(mark(X1),X2,X3))
p31: active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1))
p32: mark#(U42(X1,X2,X3)) -> mark#(X1)
p33: mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3))
p34: active#(isNat(length(V1))) -> mark#(U11(isNatIListKind(V1),V1))
p35: mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3))
p36: active#(U95(tt(),V2)) -> mark#(U96(isNatList(V2)))
p37: mark#(U33(X)) -> mark#(X)
p38: mark#(U32(X1,X2)) -> mark#(X1)
p39: mark#(U32(X1,X2)) -> active#(U32(mark(X1),X2))
p40: active#(U94(tt(),V1,V2)) -> mark#(U95(isNat(V1),V2))
p41: mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2))
p42: active#(U93(tt(),V1,V2)) -> mark#(U94(isNatIListKind(V2),V1,V2))
p43: mark#(U23(X)) -> mark#(X)
p44: mark#(U22(X1,X2)) -> mark#(X1)
p45: mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2))
p46: active#(U92(tt(),V1,V2)) -> mark#(U93(isNatIListKind(V2),V1,V2))
p47: mark#(U21(X1,X2)) -> mark#(X1)
p48: mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2))
p49: active#(U91(tt(),V1,V2)) -> mark#(U92(isNatKind(V1),V1,V2))
p50: mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2)))
p51: active#(U61(tt(),V2)) -> mark#(U62(isNatIListKind(V2)))
p52: mark#(U136(X1,X2,X3,X4)) -> active#(U136(mark(X1),X2,X3,X4))
p53: active#(U51(tt(),V2)) -> mark#(U52(isNatIListKind(V2)))
p54: mark#(U135(X1,X2,X3,X4)) -> active#(U135(mark(X1),X2,X3,X4))
p55: active#(U45(tt(),V2)) -> mark#(U46(isNatIList(V2)))
p56: mark#(U134(X1,X2,X3,X4)) -> active#(U134(mark(X1),X2,X3,X4))
p57: active#(U44(tt(),V1,V2)) -> mark#(U45(isNat(V1),V2))
p58: mark#(U133(X1,X2,X3,X4)) -> active#(U133(mark(X1),X2,X3,X4))
p59: active#(U43(tt(),V1,V2)) -> mark#(U44(isNatIListKind(V2),V1,V2))
p60: mark#(U132(X1,X2,X3,X4)) -> active#(U132(mark(X1),X2,X3,X4))
p61: active#(U42(tt(),V1,V2)) -> mark#(U43(isNatIListKind(V2),V1,V2))
p62: mark#(U131(X1,X2,X3,X4)) -> active#(U131(mark(X1),X2,X3,X4))
p63: active#(U41(tt(),V1,V2)) -> mark#(U42(isNatKind(V1),V1,V2))
p64: mark#(U122(X)) -> mark#(X)
p65: mark#(U121(X1,X2)) -> active#(U121(mark(X1),X2))
p66: active#(U32(tt(),V)) -> mark#(U33(isNatList(V)))
p67: mark#(isNatList(X)) -> active#(isNatList(X))
p68: active#(U22(tt(),V1)) -> mark#(U23(isNat(V1)))
p69: mark#(U13(X)) -> mark#(X)
p70: mark#(length(X)) -> active#(length(mark(X)))
p71: active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1))
p72: mark#(s(X)) -> mark#(X)
p73: mark#(U114(X1,X2)) -> active#(U114(mark(X1),X2))
p74: active#(U136(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL)))
p75: mark#(U113(X1,X2,X3)) -> active#(U113(mark(X1),X2,X3))
p76: active#(U135(tt(),IL,M,N)) -> mark#(U136(isNatKind(N),IL,M,N))
p77: mark#(U112(X1,X2,X3)) -> active#(U112(mark(X1),X2,X3))
p78: active#(U134(tt(),IL,M,N)) -> mark#(U135(isNat(N),IL,M,N))
p79: mark#(U111(X1,X2,X3)) -> active#(U111(mark(X1),X2,X3))
p80: active#(U133(tt(),IL,M,N)) -> mark#(U134(isNatKind(M),IL,M,N))
p81: mark#(U12(X1,X2)) -> mark#(X1)
p82: mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2))
p83: active#(U132(tt(),IL,M,N)) -> mark#(U133(isNat(M),IL,M,N))
p84: mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2))
p85: active#(U131(tt(),IL,M,N)) -> mark#(U132(isNatIListKind(IL),IL,M,N))
p86: mark#(isNatIList(X)) -> active#(isNatIList(X))
p87: active#(U121(tt(),IL)) -> mark#(U122(isNatIListKind(IL)))
p88: mark#(U106(X)) -> mark#(X)
p89: mark#(isNat(X)) -> active#(isNat(X))
p90: active#(U12(tt(),V1)) -> mark#(U13(isNatList(V1)))
p91: mark#(U105(X1,X2)) -> active#(U105(mark(X1),X2))
p92: active#(U114(tt(),L)) -> mark#(s(length(L)))
p93: mark#(U104(X1,X2,X3)) -> active#(U104(mark(X1),X2,X3))
p94: active#(U113(tt(),L,N)) -> mark#(U114(isNatKind(N),L))
p95: mark#(isNatIListKind(X)) -> active#(isNatIListKind(X))
p96: active#(U112(tt(),L,N)) -> mark#(U113(isNat(N),L,N))
p97: mark#(U103(X1,X2,X3)) -> mark#(X1)
p98: mark#(U103(X1,X2,X3)) -> active#(U103(mark(X1),X2,X3))
p99: active#(U111(tt(),L,N)) -> mark#(U112(isNatIListKind(L),L,N))
p100: mark#(isNatKind(X)) -> active#(isNatKind(X))
p101: active#(U11(tt(),V1)) -> mark#(U12(isNatIListKind(V1),V1))
p102: mark#(U102(X1,X2,X3)) -> mark#(X1)
p103: mark#(U102(X1,X2,X3)) -> active#(U102(mark(X1),X2,X3))
p104: active#(U105(tt(),V2)) -> mark#(U106(isNatIList(V2)))
p105: mark#(U101(X1,X2,X3)) -> active#(U101(mark(X1),X2,X3))
p106: active#(U104(tt(),V1,V2)) -> mark#(U105(isNat(V1),V2))
p107: active#(U103(tt(),V1,V2)) -> mark#(U104(isNatIListKind(V2),V1,V2))
p108: active#(U101(tt(),V1,V2)) -> mark#(U102(isNatKind(V1),V1,V2))

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41, r42, r43, r44, r45, r46, r47, r48, r49, r50, r51, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76, r77, r78, r79, r80, r81, r82, r83, r84, r85, r86, r87, r88, r89, r90, r91, r92, r93, r94, r95, r96, r97, r98, r99, r100, r101, r102, r103, r104, r105, r106, r107, r108, r109, r110, r111, r112, r113, r114, r115, r116, r117, r118, r119, r120, r121, r122, r123, r124, r125, r126, r127, r128, r129, r130, r131, r132, r133, r134, r135, r136, r137, r138, r139, r140, r141, r142, r143, r144, r145, r146, r147, r148, r149, r150, r151, r152, r153, r154, r155, r156, r157, r158, r159, r160, r161, r162, r163, r164, r165, r166, r167, r168, r169, r170, r171, r172, r173, r174, r175, r176, r177, r178, r179, r180, r181, r182, r183, r184, r185, r186, r187, r188, r189, r190, r191, r192, r193, r194, r195, r196, r197, r198, r199, r200, r201, r202, r203, r204, r205, r206, r207, r208, r209, r210, r211, r212, r213, r214, r215, r216, r217, r218, r219, r220, r221, r222, r223, r224, r225, r226, r227, r228, r229, r230, r231, r232, r233, r234, r235, r236, r237, r238, r239, r240, r241, r242, r243, r244, r245, r246, r247, r248, r249, r250, r251, r252, r253, r254, r255, r256, r257, r258, r259, r260, r261, r262, r263, r264, r265, r266, r267, r268, r269, r270, r271, r272, r273, r274, r275, r276, r277, r278, r279, r280, r281, r282, r283, r284, r285, r286, r287, r288, r289, r290, r291, r292, r293, r294, r295, r296, r297, r298, r299, r300, r301, r302, r303, r304, r305, r306, r307, r308, r309, r310, r311, r312, r313, r314, r315, r316, r317, r318, r319, r320, r321, r322, r323, r324, r325, r326, r327, r328, r329, r330, r331, r332, r333, r334, r335, r336, r337, r338, r339, r340, r341, r342, r343, r344, r345, r346, r347, r348, r349, r350, r351, r352, r353, r354, r355, r356, r357, r358

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    mark#_A(x1) = 414
    U96_A(x1) = 396
    U95_A(x1,x2) = 392
    active#_A(x1) = x1 + 22
    mark_A(x1) = max{284, x1 - 117}
    take_A(x1,x2) = 392
    s_A(x1) = 0
    cons_A(x1,x2) = 401
    U131_A(x1,x2,x3,x4) = 392
    isNatIList_A(x1) = 392
    U94_A(x1,x2,x3) = 392
    |0|_A = 400
    U121_A(x1,x2) = 392
    U93_A(x1,x2,x3) = 392
    length_A(x1) = 392
    U111_A(x1,x2,x3) = 392
    isNatList_A(x1) = 392
    U92_A(x1,x2,x3) = 392
    U101_A(x1,x2,x3) = 392
    isNatKind_A(x1) = 392
    U91_A(x1,x2,x3) = 392
    U81_A(x1) = 55
    U71_A(x1) = 392
    U62_A(x1) = 395
    U61_A(x1,x2) = 392
    U52_A(x1) = 401
    U51_A(x1,x2) = 392
    isNatIListKind_A(x1) = 392
    U46_A(x1) = 396
    U45_A(x1,x2) = 392
    U44_A(x1,x2,x3) = 392
    U41_A(x1,x2,x3) = 392
    U43_A(x1,x2,x3) = 392
    isNat_A(x1) = 392
    U21_A(x1,x2) = 392
    U42_A(x1,x2,x3) = 392
    U11_A(x1,x2) = 392
    tt_A = 400
    U33_A(x1) = 396
    U32_A(x1,x2) = 392
    U31_A(x1,x2) = 392
    U23_A(x1) = 396
    U22_A(x1,x2) = 392
    U136_A(x1,x2,x3,x4) = 392
    U135_A(x1,x2,x3,x4) = 392
    U134_A(x1,x2,x3,x4) = 392
    U133_A(x1,x2,x3,x4) = 392
    U132_A(x1,x2,x3,x4) = 392
    U122_A(x1) = 396
    U13_A(x1) = 396
    U114_A(x1,x2) = 392
    U113_A(x1,x2,x3) = 392
    U112_A(x1,x2,x3) = 392
    U12_A(x1,x2) = 392
    U106_A(x1) = 55
    U105_A(x1,x2) = 392
    U104_A(x1,x2,x3) = 392
    U103_A(x1,x2,x3) = 392
    U102_A(x1,x2,x3) = 0
    active_A(x1) = 284
    zeros_A = 399
    nil_A = 400

The next rules are strictly ordered:

  p103

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(U96(X)) -> mark#(X)
p2: mark#(U95(X1,X2)) -> mark#(X1)
p3: mark#(U95(X1,X2)) -> active#(U95(mark(X1),X2))
p4: active#(take(s(M),cons(N,IL))) -> mark#(U131(isNatIList(IL),IL,M,N))
p5: mark#(U94(X1,X2,X3)) -> active#(U94(mark(X1),X2,X3))
p6: active#(take(|0|(),IL)) -> mark#(U121(isNatIList(IL),IL))
p7: mark#(U93(X1,X2,X3)) -> active#(U93(mark(X1),X2,X3))
p8: active#(length(cons(N,L))) -> mark#(U111(isNatList(L),L,N))
p9: mark#(U92(X1,X2,X3)) -> mark#(X1)
p10: mark#(U92(X1,X2,X3)) -> active#(U92(mark(X1),X2,X3))
p11: active#(isNatList(take(V1,V2))) -> mark#(U101(isNatKind(V1),V1,V2))
p12: mark#(U91(X1,X2,X3)) -> mark#(X1)
p13: mark#(U91(X1,X2,X3)) -> active#(U91(mark(X1),X2,X3))
p14: active#(isNatList(cons(V1,V2))) -> mark#(U91(isNatKind(V1),V1,V2))
p15: mark#(U81(X)) -> mark#(X)
p16: mark#(U71(X)) -> mark#(X)
p17: mark#(U62(X)) -> mark#(X)
p18: mark#(U61(X1,X2)) -> active#(U61(mark(X1),X2))
p19: active#(isNatKind(s(V1))) -> mark#(U81(isNatKind(V1)))
p20: mark#(U52(X)) -> mark#(X)
p21: mark#(U51(X1,X2)) -> mark#(X1)
p22: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p23: active#(isNatKind(length(V1))) -> mark#(U71(isNatIListKind(V1)))
p24: mark#(U46(X)) -> mark#(X)
p25: mark#(U45(X1,X2)) -> mark#(X1)
p26: mark#(U45(X1,X2)) -> active#(U45(mark(X1),X2))
p27: active#(isNatIListKind(cons(V1,V2))) -> mark#(U51(isNatKind(V1),V2))
p28: mark#(U44(X1,X2,X3)) -> active#(U44(mark(X1),X2,X3))
p29: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNatKind(V1),V1,V2))
p30: mark#(U43(X1,X2,X3)) -> active#(U43(mark(X1),X2,X3))
p31: active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1))
p32: mark#(U42(X1,X2,X3)) -> mark#(X1)
p33: mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3))
p34: active#(isNat(length(V1))) -> mark#(U11(isNatIListKind(V1),V1))
p35: mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3))
p36: active#(U95(tt(),V2)) -> mark#(U96(isNatList(V2)))
p37: mark#(U33(X)) -> mark#(X)
p38: mark#(U32(X1,X2)) -> mark#(X1)
p39: mark#(U32(X1,X2)) -> active#(U32(mark(X1),X2))
p40: active#(U94(tt(),V1,V2)) -> mark#(U95(isNat(V1),V2))
p41: mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2))
p42: active#(U93(tt(),V1,V2)) -> mark#(U94(isNatIListKind(V2),V1,V2))
p43: mark#(U23(X)) -> mark#(X)
p44: mark#(U22(X1,X2)) -> mark#(X1)
p45: mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2))
p46: active#(U92(tt(),V1,V2)) -> mark#(U93(isNatIListKind(V2),V1,V2))
p47: mark#(U21(X1,X2)) -> mark#(X1)
p48: mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2))
p49: active#(U91(tt(),V1,V2)) -> mark#(U92(isNatKind(V1),V1,V2))
p50: mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2)))
p51: active#(U61(tt(),V2)) -> mark#(U62(isNatIListKind(V2)))
p52: mark#(U136(X1,X2,X3,X4)) -> active#(U136(mark(X1),X2,X3,X4))
p53: active#(U51(tt(),V2)) -> mark#(U52(isNatIListKind(V2)))
p54: mark#(U135(X1,X2,X3,X4)) -> active#(U135(mark(X1),X2,X3,X4))
p55: active#(U45(tt(),V2)) -> mark#(U46(isNatIList(V2)))
p56: mark#(U134(X1,X2,X3,X4)) -> active#(U134(mark(X1),X2,X3,X4))
p57: active#(U44(tt(),V1,V2)) -> mark#(U45(isNat(V1),V2))
p58: mark#(U133(X1,X2,X3,X4)) -> active#(U133(mark(X1),X2,X3,X4))
p59: active#(U43(tt(),V1,V2)) -> mark#(U44(isNatIListKind(V2),V1,V2))
p60: mark#(U132(X1,X2,X3,X4)) -> active#(U132(mark(X1),X2,X3,X4))
p61: active#(U42(tt(),V1,V2)) -> mark#(U43(isNatIListKind(V2),V1,V2))
p62: mark#(U131(X1,X2,X3,X4)) -> active#(U131(mark(X1),X2,X3,X4))
p63: active#(U41(tt(),V1,V2)) -> mark#(U42(isNatKind(V1),V1,V2))
p64: mark#(U122(X)) -> mark#(X)
p65: mark#(U121(X1,X2)) -> active#(U121(mark(X1),X2))
p66: active#(U32(tt(),V)) -> mark#(U33(isNatList(V)))
p67: mark#(isNatList(X)) -> active#(isNatList(X))
p68: active#(U22(tt(),V1)) -> mark#(U23(isNat(V1)))
p69: mark#(U13(X)) -> mark#(X)
p70: mark#(length(X)) -> active#(length(mark(X)))
p71: active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1))
p72: mark#(s(X)) -> mark#(X)
p73: mark#(U114(X1,X2)) -> active#(U114(mark(X1),X2))
p74: active#(U136(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL)))
p75: mark#(U113(X1,X2,X3)) -> active#(U113(mark(X1),X2,X3))
p76: active#(U135(tt(),IL,M,N)) -> mark#(U136(isNatKind(N),IL,M,N))
p77: mark#(U112(X1,X2,X3)) -> active#(U112(mark(X1),X2,X3))
p78: active#(U134(tt(),IL,M,N)) -> mark#(U135(isNat(N),IL,M,N))
p79: mark#(U111(X1,X2,X3)) -> active#(U111(mark(X1),X2,X3))
p80: active#(U133(tt(),IL,M,N)) -> mark#(U134(isNatKind(M),IL,M,N))
p81: mark#(U12(X1,X2)) -> mark#(X1)
p82: mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2))
p83: active#(U132(tt(),IL,M,N)) -> mark#(U133(isNat(M),IL,M,N))
p84: mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2))
p85: active#(U131(tt(),IL,M,N)) -> mark#(U132(isNatIListKind(IL),IL,M,N))
p86: mark#(isNatIList(X)) -> active#(isNatIList(X))
p87: active#(U121(tt(),IL)) -> mark#(U122(isNatIListKind(IL)))
p88: mark#(U106(X)) -> mark#(X)
p89: mark#(isNat(X)) -> active#(isNat(X))
p90: active#(U12(tt(),V1)) -> mark#(U13(isNatList(V1)))
p91: mark#(U105(X1,X2)) -> active#(U105(mark(X1),X2))
p92: active#(U114(tt(),L)) -> mark#(s(length(L)))
p93: mark#(U104(X1,X2,X3)) -> active#(U104(mark(X1),X2,X3))
p94: active#(U113(tt(),L,N)) -> mark#(U114(isNatKind(N),L))
p95: mark#(isNatIListKind(X)) -> active#(isNatIListKind(X))
p96: active#(U112(tt(),L,N)) -> mark#(U113(isNat(N),L,N))
p97: mark#(U103(X1,X2,X3)) -> mark#(X1)
p98: mark#(U103(X1,X2,X3)) -> active#(U103(mark(X1),X2,X3))
p99: active#(U111(tt(),L,N)) -> mark#(U112(isNatIListKind(L),L,N))
p100: mark#(isNatKind(X)) -> active#(isNatKind(X))
p101: active#(U11(tt(),V1)) -> mark#(U12(isNatIListKind(V1),V1))
p102: mark#(U102(X1,X2,X3)) -> mark#(X1)
p103: active#(U105(tt(),V2)) -> mark#(U106(isNatIList(V2)))
p104: mark#(U101(X1,X2,X3)) -> active#(U101(mark(X1),X2,X3))
p105: active#(U104(tt(),V1,V2)) -> mark#(U105(isNat(V1),V2))
p106: active#(U103(tt(),V1,V2)) -> mark#(U104(isNatIListKind(V2),V1,V2))
p107: active#(U101(tt(),V1,V2)) -> mark#(U102(isNatKind(V1),V1,V2))

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19, p20, p21, p22, p23, p24, p25, p26, p27, p28, p29, p30, p31, p32, p33, p34, p35, p36, p37, p38, p39, p40, p41, p42, p43, p44, p45, p46, p47, p48, p49, p50, p51, p52, p53, p54, p55, p56, p57, p58, p59, p60, p61, p62, p63, p64, p65, p66, p67, p68, p69, p70, p71, p72, p73, p74, p75, p76, p77, p78, p79, p80, p81, p82, p83, p84, p85, p86, p87, p88, p89, p90, p91, p92, p93, p94, p95, p96, p97, p98, p99, p100, p101, p102, p103, p104, p105, p106, p107}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(U96(X)) -> mark#(X)
p2: mark#(U101(X1,X2,X3)) -> active#(U101(mark(X1),X2,X3))
p3: active#(U101(tt(),V1,V2)) -> mark#(U102(isNatKind(V1),V1,V2))
p4: mark#(U102(X1,X2,X3)) -> mark#(X1)
p5: mark#(isNatKind(X)) -> active#(isNatKind(X))
p6: active#(U103(tt(),V1,V2)) -> mark#(U104(isNatIListKind(V2),V1,V2))
p7: mark#(U103(X1,X2,X3)) -> active#(U103(mark(X1),X2,X3))
p8: active#(U104(tt(),V1,V2)) -> mark#(U105(isNat(V1),V2))
p9: mark#(U103(X1,X2,X3)) -> mark#(X1)
p10: mark#(isNatIListKind(X)) -> active#(isNatIListKind(X))
p11: active#(U105(tt(),V2)) -> mark#(U106(isNatIList(V2)))
p12: mark#(U104(X1,X2,X3)) -> active#(U104(mark(X1),X2,X3))
p13: active#(U11(tt(),V1)) -> mark#(U12(isNatIListKind(V1),V1))
p14: mark#(U105(X1,X2)) -> active#(U105(mark(X1),X2))
p15: active#(U111(tt(),L,N)) -> mark#(U112(isNatIListKind(L),L,N))
p16: mark#(isNat(X)) -> active#(isNat(X))
p17: active#(U112(tt(),L,N)) -> mark#(U113(isNat(N),L,N))
p18: mark#(U106(X)) -> mark#(X)
p19: mark#(isNatIList(X)) -> active#(isNatIList(X))
p20: active#(U113(tt(),L,N)) -> mark#(U114(isNatKind(N),L))
p21: mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2))
p22: active#(U114(tt(),L)) -> mark#(s(length(L)))
p23: mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2))
p24: active#(U12(tt(),V1)) -> mark#(U13(isNatList(V1)))
p25: mark#(U12(X1,X2)) -> mark#(X1)
p26: mark#(U111(X1,X2,X3)) -> active#(U111(mark(X1),X2,X3))
p27: active#(U121(tt(),IL)) -> mark#(U122(isNatIListKind(IL)))
p28: mark#(U112(X1,X2,X3)) -> active#(U112(mark(X1),X2,X3))
p29: active#(U131(tt(),IL,M,N)) -> mark#(U132(isNatIListKind(IL),IL,M,N))
p30: mark#(U113(X1,X2,X3)) -> active#(U113(mark(X1),X2,X3))
p31: active#(U132(tt(),IL,M,N)) -> mark#(U133(isNat(M),IL,M,N))
p32: mark#(U114(X1,X2)) -> active#(U114(mark(X1),X2))
p33: active#(U133(tt(),IL,M,N)) -> mark#(U134(isNatKind(M),IL,M,N))
p34: mark#(s(X)) -> mark#(X)
p35: mark#(length(X)) -> active#(length(mark(X)))
p36: active#(U134(tt(),IL,M,N)) -> mark#(U135(isNat(N),IL,M,N))
p37: mark#(U13(X)) -> mark#(X)
p38: mark#(isNatList(X)) -> active#(isNatList(X))
p39: active#(U135(tt(),IL,M,N)) -> mark#(U136(isNatKind(N),IL,M,N))
p40: mark#(U121(X1,X2)) -> active#(U121(mark(X1),X2))
p41: active#(U136(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL)))
p42: mark#(U122(X)) -> mark#(X)
p43: mark#(U131(X1,X2,X3,X4)) -> active#(U131(mark(X1),X2,X3,X4))
p44: active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1))
p45: mark#(U132(X1,X2,X3,X4)) -> active#(U132(mark(X1),X2,X3,X4))
p46: active#(U22(tt(),V1)) -> mark#(U23(isNat(V1)))
p47: mark#(U133(X1,X2,X3,X4)) -> active#(U133(mark(X1),X2,X3,X4))
p48: active#(U32(tt(),V)) -> mark#(U33(isNatList(V)))
p49: mark#(U134(X1,X2,X3,X4)) -> active#(U134(mark(X1),X2,X3,X4))
p50: active#(U41(tt(),V1,V2)) -> mark#(U42(isNatKind(V1),V1,V2))
p51: mark#(U135(X1,X2,X3,X4)) -> active#(U135(mark(X1),X2,X3,X4))
p52: active#(U42(tt(),V1,V2)) -> mark#(U43(isNatIListKind(V2),V1,V2))
p53: mark#(U136(X1,X2,X3,X4)) -> active#(U136(mark(X1),X2,X3,X4))
p54: active#(U43(tt(),V1,V2)) -> mark#(U44(isNatIListKind(V2),V1,V2))
p55: mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2)))
p56: active#(U44(tt(),V1,V2)) -> mark#(U45(isNat(V1),V2))
p57: mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2))
p58: active#(U45(tt(),V2)) -> mark#(U46(isNatIList(V2)))
p59: mark#(U21(X1,X2)) -> mark#(X1)
p60: mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2))
p61: active#(U51(tt(),V2)) -> mark#(U52(isNatIListKind(V2)))
p62: mark#(U22(X1,X2)) -> mark#(X1)
p63: mark#(U23(X)) -> mark#(X)
p64: mark#(U31(X1,X2)) -> active#(U31(mark(X1),X2))
p65: active#(U61(tt(),V2)) -> mark#(U62(isNatIListKind(V2)))
p66: mark#(U32(X1,X2)) -> active#(U32(mark(X1),X2))
p67: active#(U91(tt(),V1,V2)) -> mark#(U92(isNatKind(V1),V1,V2))
p68: mark#(U32(X1,X2)) -> mark#(X1)
p69: mark#(U33(X)) -> mark#(X)
p70: mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3))
p71: active#(U92(tt(),V1,V2)) -> mark#(U93(isNatIListKind(V2),V1,V2))
p72: mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3))
p73: active#(U93(tt(),V1,V2)) -> mark#(U94(isNatIListKind(V2),V1,V2))
p74: mark#(U42(X1,X2,X3)) -> mark#(X1)
p75: mark#(U43(X1,X2,X3)) -> active#(U43(mark(X1),X2,X3))
p76: active#(U94(tt(),V1,V2)) -> mark#(U95(isNat(V1),V2))
p77: mark#(U44(X1,X2,X3)) -> active#(U44(mark(X1),X2,X3))
p78: active#(U95(tt(),V2)) -> mark#(U96(isNatList(V2)))
p79: mark#(U45(X1,X2)) -> active#(U45(mark(X1),X2))
p80: active#(isNat(length(V1))) -> mark#(U11(isNatIListKind(V1),V1))
p81: mark#(U45(X1,X2)) -> mark#(X1)
p82: mark#(U46(X)) -> mark#(X)
p83: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p84: active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1))
p85: mark#(U51(X1,X2)) -> mark#(X1)
p86: mark#(U52(X)) -> mark#(X)
p87: mark#(U61(X1,X2)) -> active#(U61(mark(X1),X2))
p88: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNatKind(V1),V1,V2))
p89: mark#(U62(X)) -> mark#(X)
p90: mark#(U71(X)) -> mark#(X)
p91: mark#(U81(X)) -> mark#(X)
p92: mark#(U91(X1,X2,X3)) -> active#(U91(mark(X1),X2,X3))
p93: active#(isNatIListKind(cons(V1,V2))) -> mark#(U51(isNatKind(V1),V2))
p94: mark#(U91(X1,X2,X3)) -> mark#(X1)
p95: mark#(U92(X1,X2,X3)) -> active#(U92(mark(X1),X2,X3))
p96: active#(isNatKind(length(V1))) -> mark#(U71(isNatIListKind(V1)))
p97: mark#(U92(X1,X2,X3)) -> mark#(X1)
p98: mark#(U93(X1,X2,X3)) -> active#(U93(mark(X1),X2,X3))
p99: active#(isNatKind(s(V1))) -> mark#(U81(isNatKind(V1)))
p100: mark#(U94(X1,X2,X3)) -> active#(U94(mark(X1),X2,X3))
p101: active#(isNatList(cons(V1,V2))) -> mark#(U91(isNatKind(V1),V1,V2))
p102: mark#(U95(X1,X2)) -> active#(U95(mark(X1),X2))
p103: active#(isNatList(take(V1,V2))) -> mark#(U101(isNatKind(V1),V1,V2))
p104: mark#(U95(X1,X2)) -> mark#(X1)
p105: active#(length(cons(N,L))) -> mark#(U111(isNatList(L),L,N))
p106: active#(take(|0|(),IL)) -> mark#(U121(isNatIList(IL),IL))
p107: active#(take(s(M),cons(N,IL))) -> mark#(U131(isNatIList(IL),IL,M,N))

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41, r42, r43, r44, r45, r46, r47, r48, r49, r50, r51, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76, r77, r78, r79, r80, r81, r82, r83, r84, r85, r86, r87, r88, r89, r90, r91, r92, r93, r94, r95, r96, r97, r98, r99, r100, r101, r102, r103, r104, r105, r106, r107, r108, r109, r110, r111, r112, r113, r114, r115, r116, r117, r118, r119, r120, r121, r122, r123, r124, r125, r126, r127, r128, r129, r130, r131, r132, r133, r134, r135, r136, r137, r138, r139, r140, r141, r142, r143, r144, r145, r146, r147, r148, r149, r150, r151, r152, r153, r154, r155, r156, r157, r158, r159, r160, r161, r162, r163, r164, r165, r166, r167, r168, r169, r170, r171, r172, r173, r174, r175, r176, r177, r178, r179, r180, r181, r182, r183, r184, r185, r186, r187, r188, r189, r190, r191, r192, r193, r194, r195, r196, r197, r198, r199, r200, r201, r202, r203, r204, r205, r206, r207, r208, r209, r210, r211, r212, r213, r214, r215, r216, r217, r218, r219, r220, r221, r222, r223, r224, r225, r226, r227, r228, r229, r230, r231, r232, r233, r234, r235, r236, r237, r238, r239, r240, r241, r242, r243, r244, r245, r246, r247, r248, r249, r250, r251, r252, r253, r254, r255, r256, r257, r258, r259, r260, r261, r262, r263, r264, r265, r266, r267, r268, r269, r270, r271, r272, r273, r274, r275, r276, r277, r278, r279, r280, r281, r282, r283, r284, r285, r286, r287, r288, r289, r290, r291, r292, r293, r294, r295, r296, r297, r298, r299, r300, r301, r302, r303, r304, r305, r306, r307, r308, r309, r310, r311, r312, r313, r314, r315, r316, r317, r318, r319, r320, r321, r322, r323, r324, r325, r326, r327, r328, r329, r330, r331, r332, r333, r334, r335, r336, r337, r338, r339, r340, r341, r342, r343, r344, r345, r346, r347, r348, r349, r350, r351, r352, r353, r354, r355, r356, r357, r358

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    mark#_A(x1) = 23
    U96_A(x1) = 56
    U101_A(x1,x2,x3) = 29
    active#_A(x1) = max{0, x1 - 6}
    mark_A(x1) = max{55, x1 - 1}
    tt_A = 27
    U102_A(x1,x2,x3) = 29
    isNatKind_A(x1) = 29
    U103_A(x1,x2,x3) = 29
    U104_A(x1,x2,x3) = 29
    isNatIListKind_A(x1) = 29
    U105_A(x1,x2) = 29
    isNat_A(x1) = 29
    U106_A(x1) = 56
    isNatIList_A(x1) = 29
    U11_A(x1,x2) = 29
    U12_A(x1,x2) = 29
    U111_A(x1,x2,x3) = 29
    U112_A(x1,x2,x3) = 29
    U113_A(x1,x2,x3) = 29
    U114_A(x1,x2) = 29
    s_A(x1) = 0
    length_A(x1) = 29
    U13_A(x1) = 56
    isNatList_A(x1) = 29
    U121_A(x1,x2) = 29
    U122_A(x1) = 30
    U131_A(x1,x2,x3,x4) = 29
    U132_A(x1,x2,x3,x4) = 29
    U133_A(x1,x2,x3,x4) = 29
    U134_A(x1,x2,x3,x4) = 29
    U135_A(x1,x2,x3,x4) = 29
    U136_A(x1,x2,x3,x4) = 29
    cons_A(x1,x2) = 20
    take_A(x1,x2) = 29
    U21_A(x1,x2) = 29
    U22_A(x1,x2) = 29
    U23_A(x1) = 29
    U32_A(x1,x2) = 29
    U33_A(x1) = 56
    U41_A(x1,x2,x3) = 29
    U42_A(x1,x2,x3) = 29
    U43_A(x1,x2,x3) = 29
    U44_A(x1,x2,x3) = 29
    U45_A(x1,x2) = 29
    U46_A(x1) = 56
    U51_A(x1,x2) = 29
    U52_A(x1) = 56
    U31_A(x1,x2) = 28
    U61_A(x1,x2) = 29
    U62_A(x1) = 56
    U91_A(x1,x2,x3) = 29
    U92_A(x1,x2,x3) = 29
    U93_A(x1,x2,x3) = 29
    U94_A(x1,x2,x3) = 29
    U95_A(x1,x2) = 29
    U71_A(x1) = 56
    U81_A(x1) = 29
    |0|_A = 56
    active_A(x1) = max{55, x1 - 1}
    zeros_A = 65
    nil_A = 55

The next rules are strictly ordered:

  p64

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(U96(X)) -> mark#(X)
p2: mark#(U101(X1,X2,X3)) -> active#(U101(mark(X1),X2,X3))
p3: active#(U101(tt(),V1,V2)) -> mark#(U102(isNatKind(V1),V1,V2))
p4: mark#(U102(X1,X2,X3)) -> mark#(X1)
p5: mark#(isNatKind(X)) -> active#(isNatKind(X))
p6: active#(U103(tt(),V1,V2)) -> mark#(U104(isNatIListKind(V2),V1,V2))
p7: mark#(U103(X1,X2,X3)) -> active#(U103(mark(X1),X2,X3))
p8: active#(U104(tt(),V1,V2)) -> mark#(U105(isNat(V1),V2))
p9: mark#(U103(X1,X2,X3)) -> mark#(X1)
p10: mark#(isNatIListKind(X)) -> active#(isNatIListKind(X))
p11: active#(U105(tt(),V2)) -> mark#(U106(isNatIList(V2)))
p12: mark#(U104(X1,X2,X3)) -> active#(U104(mark(X1),X2,X3))
p13: active#(U11(tt(),V1)) -> mark#(U12(isNatIListKind(V1),V1))
p14: mark#(U105(X1,X2)) -> active#(U105(mark(X1),X2))
p15: active#(U111(tt(),L,N)) -> mark#(U112(isNatIListKind(L),L,N))
p16: mark#(isNat(X)) -> active#(isNat(X))
p17: active#(U112(tt(),L,N)) -> mark#(U113(isNat(N),L,N))
p18: mark#(U106(X)) -> mark#(X)
p19: mark#(isNatIList(X)) -> active#(isNatIList(X))
p20: active#(U113(tt(),L,N)) -> mark#(U114(isNatKind(N),L))
p21: mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2))
p22: active#(U114(tt(),L)) -> mark#(s(length(L)))
p23: mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2))
p24: active#(U12(tt(),V1)) -> mark#(U13(isNatList(V1)))
p25: mark#(U12(X1,X2)) -> mark#(X1)
p26: mark#(U111(X1,X2,X3)) -> active#(U111(mark(X1),X2,X3))
p27: active#(U121(tt(),IL)) -> mark#(U122(isNatIListKind(IL)))
p28: mark#(U112(X1,X2,X3)) -> active#(U112(mark(X1),X2,X3))
p29: active#(U131(tt(),IL,M,N)) -> mark#(U132(isNatIListKind(IL),IL,M,N))
p30: mark#(U113(X1,X2,X3)) -> active#(U113(mark(X1),X2,X3))
p31: active#(U132(tt(),IL,M,N)) -> mark#(U133(isNat(M),IL,M,N))
p32: mark#(U114(X1,X2)) -> active#(U114(mark(X1),X2))
p33: active#(U133(tt(),IL,M,N)) -> mark#(U134(isNatKind(M),IL,M,N))
p34: mark#(s(X)) -> mark#(X)
p35: mark#(length(X)) -> active#(length(mark(X)))
p36: active#(U134(tt(),IL,M,N)) -> mark#(U135(isNat(N),IL,M,N))
p37: mark#(U13(X)) -> mark#(X)
p38: mark#(isNatList(X)) -> active#(isNatList(X))
p39: active#(U135(tt(),IL,M,N)) -> mark#(U136(isNatKind(N),IL,M,N))
p40: mark#(U121(X1,X2)) -> active#(U121(mark(X1),X2))
p41: active#(U136(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL)))
p42: mark#(U122(X)) -> mark#(X)
p43: mark#(U131(X1,X2,X3,X4)) -> active#(U131(mark(X1),X2,X3,X4))
p44: active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1))
p45: mark#(U132(X1,X2,X3,X4)) -> active#(U132(mark(X1),X2,X3,X4))
p46: active#(U22(tt(),V1)) -> mark#(U23(isNat(V1)))
p47: mark#(U133(X1,X2,X3,X4)) -> active#(U133(mark(X1),X2,X3,X4))
p48: active#(U32(tt(),V)) -> mark#(U33(isNatList(V)))
p49: mark#(U134(X1,X2,X3,X4)) -> active#(U134(mark(X1),X2,X3,X4))
p50: active#(U41(tt(),V1,V2)) -> mark#(U42(isNatKind(V1),V1,V2))
p51: mark#(U135(X1,X2,X3,X4)) -> active#(U135(mark(X1),X2,X3,X4))
p52: active#(U42(tt(),V1,V2)) -> mark#(U43(isNatIListKind(V2),V1,V2))
p53: mark#(U136(X1,X2,X3,X4)) -> active#(U136(mark(X1),X2,X3,X4))
p54: active#(U43(tt(),V1,V2)) -> mark#(U44(isNatIListKind(V2),V1,V2))
p55: mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2)))
p56: active#(U44(tt(),V1,V2)) -> mark#(U45(isNat(V1),V2))
p57: mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2))
p58: active#(U45(tt(),V2)) -> mark#(U46(isNatIList(V2)))
p59: mark#(U21(X1,X2)) -> mark#(X1)
p60: mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2))
p61: active#(U51(tt(),V2)) -> mark#(U52(isNatIListKind(V2)))
p62: mark#(U22(X1,X2)) -> mark#(X1)
p63: mark#(U23(X)) -> mark#(X)
p64: active#(U61(tt(),V2)) -> mark#(U62(isNatIListKind(V2)))
p65: mark#(U32(X1,X2)) -> active#(U32(mark(X1),X2))
p66: active#(U91(tt(),V1,V2)) -> mark#(U92(isNatKind(V1),V1,V2))
p67: mark#(U32(X1,X2)) -> mark#(X1)
p68: mark#(U33(X)) -> mark#(X)
p69: mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3))
p70: active#(U92(tt(),V1,V2)) -> mark#(U93(isNatIListKind(V2),V1,V2))
p71: mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3))
p72: active#(U93(tt(),V1,V2)) -> mark#(U94(isNatIListKind(V2),V1,V2))
p73: mark#(U42(X1,X2,X3)) -> mark#(X1)
p74: mark#(U43(X1,X2,X3)) -> active#(U43(mark(X1),X2,X3))
p75: active#(U94(tt(),V1,V2)) -> mark#(U95(isNat(V1),V2))
p76: mark#(U44(X1,X2,X3)) -> active#(U44(mark(X1),X2,X3))
p77: active#(U95(tt(),V2)) -> mark#(U96(isNatList(V2)))
p78: mark#(U45(X1,X2)) -> active#(U45(mark(X1),X2))
p79: active#(isNat(length(V1))) -> mark#(U11(isNatIListKind(V1),V1))
p80: mark#(U45(X1,X2)) -> mark#(X1)
p81: mark#(U46(X)) -> mark#(X)
p82: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p83: active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1))
p84: mark#(U51(X1,X2)) -> mark#(X1)
p85: mark#(U52(X)) -> mark#(X)
p86: mark#(U61(X1,X2)) -> active#(U61(mark(X1),X2))
p87: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNatKind(V1),V1,V2))
p88: mark#(U62(X)) -> mark#(X)
p89: mark#(U71(X)) -> mark#(X)
p90: mark#(U81(X)) -> mark#(X)
p91: mark#(U91(X1,X2,X3)) -> active#(U91(mark(X1),X2,X3))
p92: active#(isNatIListKind(cons(V1,V2))) -> mark#(U51(isNatKind(V1),V2))
p93: mark#(U91(X1,X2,X3)) -> mark#(X1)
p94: mark#(U92(X1,X2,X3)) -> active#(U92(mark(X1),X2,X3))
p95: active#(isNatKind(length(V1))) -> mark#(U71(isNatIListKind(V1)))
p96: mark#(U92(X1,X2,X3)) -> mark#(X1)
p97: mark#(U93(X1,X2,X3)) -> active#(U93(mark(X1),X2,X3))
p98: active#(isNatKind(s(V1))) -> mark#(U81(isNatKind(V1)))
p99: mark#(U94(X1,X2,X3)) -> active#(U94(mark(X1),X2,X3))
p100: active#(isNatList(cons(V1,V2))) -> mark#(U91(isNatKind(V1),V1,V2))
p101: mark#(U95(X1,X2)) -> active#(U95(mark(X1),X2))
p102: active#(isNatList(take(V1,V2))) -> mark#(U101(isNatKind(V1),V1,V2))
p103: mark#(U95(X1,X2)) -> mark#(X1)
p104: active#(length(cons(N,L))) -> mark#(U111(isNatList(L),L,N))
p105: active#(take(|0|(),IL)) -> mark#(U121(isNatIList(IL),IL))
p106: active#(take(s(M),cons(N,IL))) -> mark#(U131(isNatIList(IL),IL,M,N))

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19, p20, p21, p22, p23, p24, p25, p26, p27, p28, p29, p30, p31, p32, p33, p34, p35, p36, p37, p38, p39, p40, p41, p42, p43, p44, p45, p46, p47, p48, p49, p50, p51, p52, p53, p54, p55, p56, p57, p58, p59, p60, p61, p62, p63, p64, p65, p66, p67, p68, p69, p70, p71, p72, p73, p74, p75, p76, p77, p78, p79, p80, p81, p82, p83, p84, p85, p86, p87, p88, p89, p90, p91, p92, p93, p94, p95, p96, p97, p98, p99, p100, p101, p102, p103, p104, p105, p106}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(U96(X)) -> mark#(X)
p2: mark#(U95(X1,X2)) -> mark#(X1)
p3: mark#(U95(X1,X2)) -> active#(U95(mark(X1),X2))
p4: active#(take(s(M),cons(N,IL))) -> mark#(U131(isNatIList(IL),IL,M,N))
p5: mark#(U94(X1,X2,X3)) -> active#(U94(mark(X1),X2,X3))
p6: active#(take(|0|(),IL)) -> mark#(U121(isNatIList(IL),IL))
p7: mark#(U93(X1,X2,X3)) -> active#(U93(mark(X1),X2,X3))
p8: active#(length(cons(N,L))) -> mark#(U111(isNatList(L),L,N))
p9: mark#(U92(X1,X2,X3)) -> mark#(X1)
p10: mark#(U92(X1,X2,X3)) -> active#(U92(mark(X1),X2,X3))
p11: active#(isNatList(take(V1,V2))) -> mark#(U101(isNatKind(V1),V1,V2))
p12: mark#(U91(X1,X2,X3)) -> mark#(X1)
p13: mark#(U91(X1,X2,X3)) -> active#(U91(mark(X1),X2,X3))
p14: active#(isNatList(cons(V1,V2))) -> mark#(U91(isNatKind(V1),V1,V2))
p15: mark#(U81(X)) -> mark#(X)
p16: mark#(U71(X)) -> mark#(X)
p17: mark#(U62(X)) -> mark#(X)
p18: mark#(U61(X1,X2)) -> active#(U61(mark(X1),X2))
p19: active#(isNatKind(s(V1))) -> mark#(U81(isNatKind(V1)))
p20: mark#(U52(X)) -> mark#(X)
p21: mark#(U51(X1,X2)) -> mark#(X1)
p22: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p23: active#(isNatKind(length(V1))) -> mark#(U71(isNatIListKind(V1)))
p24: mark#(U46(X)) -> mark#(X)
p25: mark#(U45(X1,X2)) -> mark#(X1)
p26: mark#(U45(X1,X2)) -> active#(U45(mark(X1),X2))
p27: active#(isNatIListKind(cons(V1,V2))) -> mark#(U51(isNatKind(V1),V2))
p28: mark#(U44(X1,X2,X3)) -> active#(U44(mark(X1),X2,X3))
p29: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNatKind(V1),V1,V2))
p30: mark#(U43(X1,X2,X3)) -> active#(U43(mark(X1),X2,X3))
p31: active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1))
p32: mark#(U42(X1,X2,X3)) -> mark#(X1)
p33: mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3))
p34: active#(isNat(length(V1))) -> mark#(U11(isNatIListKind(V1),V1))
p35: mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3))
p36: active#(U95(tt(),V2)) -> mark#(U96(isNatList(V2)))
p37: mark#(U33(X)) -> mark#(X)
p38: mark#(U32(X1,X2)) -> mark#(X1)
p39: mark#(U32(X1,X2)) -> active#(U32(mark(X1),X2))
p40: active#(U94(tt(),V1,V2)) -> mark#(U95(isNat(V1),V2))
p41: mark#(U23(X)) -> mark#(X)
p42: mark#(U22(X1,X2)) -> mark#(X1)
p43: mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2))
p44: active#(U93(tt(),V1,V2)) -> mark#(U94(isNatIListKind(V2),V1,V2))
p45: mark#(U21(X1,X2)) -> mark#(X1)
p46: mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2))
p47: active#(U92(tt(),V1,V2)) -> mark#(U93(isNatIListKind(V2),V1,V2))
p48: mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2)))
p49: active#(U91(tt(),V1,V2)) -> mark#(U92(isNatKind(V1),V1,V2))
p50: mark#(U136(X1,X2,X3,X4)) -> active#(U136(mark(X1),X2,X3,X4))
p51: active#(U61(tt(),V2)) -> mark#(U62(isNatIListKind(V2)))
p52: mark#(U135(X1,X2,X3,X4)) -> active#(U135(mark(X1),X2,X3,X4))
p53: active#(U51(tt(),V2)) -> mark#(U52(isNatIListKind(V2)))
p54: mark#(U134(X1,X2,X3,X4)) -> active#(U134(mark(X1),X2,X3,X4))
p55: active#(U45(tt(),V2)) -> mark#(U46(isNatIList(V2)))
p56: mark#(U133(X1,X2,X3,X4)) -> active#(U133(mark(X1),X2,X3,X4))
p57: active#(U44(tt(),V1,V2)) -> mark#(U45(isNat(V1),V2))
p58: mark#(U132(X1,X2,X3,X4)) -> active#(U132(mark(X1),X2,X3,X4))
p59: active#(U43(tt(),V1,V2)) -> mark#(U44(isNatIListKind(V2),V1,V2))
p60: mark#(U131(X1,X2,X3,X4)) -> active#(U131(mark(X1),X2,X3,X4))
p61: active#(U42(tt(),V1,V2)) -> mark#(U43(isNatIListKind(V2),V1,V2))
p62: mark#(U122(X)) -> mark#(X)
p63: mark#(U121(X1,X2)) -> active#(U121(mark(X1),X2))
p64: active#(U41(tt(),V1,V2)) -> mark#(U42(isNatKind(V1),V1,V2))
p65: mark#(isNatList(X)) -> active#(isNatList(X))
p66: active#(U32(tt(),V)) -> mark#(U33(isNatList(V)))
p67: mark#(U13(X)) -> mark#(X)
p68: mark#(length(X)) -> active#(length(mark(X)))
p69: active#(U22(tt(),V1)) -> mark#(U23(isNat(V1)))
p70: mark#(s(X)) -> mark#(X)
p71: mark#(U114(X1,X2)) -> active#(U114(mark(X1),X2))
p72: active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1))
p73: mark#(U113(X1,X2,X3)) -> active#(U113(mark(X1),X2,X3))
p74: active#(U136(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL)))
p75: mark#(U112(X1,X2,X3)) -> active#(U112(mark(X1),X2,X3))
p76: active#(U135(tt(),IL,M,N)) -> mark#(U136(isNatKind(N),IL,M,N))
p77: mark#(U111(X1,X2,X3)) -> active#(U111(mark(X1),X2,X3))
p78: active#(U134(tt(),IL,M,N)) -> mark#(U135(isNat(N),IL,M,N))
p79: mark#(U12(X1,X2)) -> mark#(X1)
p80: mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2))
p81: active#(U133(tt(),IL,M,N)) -> mark#(U134(isNatKind(M),IL,M,N))
p82: mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2))
p83: active#(U132(tt(),IL,M,N)) -> mark#(U133(isNat(M),IL,M,N))
p84: mark#(isNatIList(X)) -> active#(isNatIList(X))
p85: active#(U131(tt(),IL,M,N)) -> mark#(U132(isNatIListKind(IL),IL,M,N))
p86: mark#(U106(X)) -> mark#(X)
p87: mark#(isNat(X)) -> active#(isNat(X))
p88: active#(U121(tt(),IL)) -> mark#(U122(isNatIListKind(IL)))
p89: mark#(U105(X1,X2)) -> active#(U105(mark(X1),X2))
p90: active#(U12(tt(),V1)) -> mark#(U13(isNatList(V1)))
p91: mark#(U104(X1,X2,X3)) -> active#(U104(mark(X1),X2,X3))
p92: active#(U114(tt(),L)) -> mark#(s(length(L)))
p93: mark#(isNatIListKind(X)) -> active#(isNatIListKind(X))
p94: active#(U113(tt(),L,N)) -> mark#(U114(isNatKind(N),L))
p95: mark#(U103(X1,X2,X3)) -> mark#(X1)
p96: mark#(U103(X1,X2,X3)) -> active#(U103(mark(X1),X2,X3))
p97: active#(U112(tt(),L,N)) -> mark#(U113(isNat(N),L,N))
p98: mark#(isNatKind(X)) -> active#(isNatKind(X))
p99: active#(U111(tt(),L,N)) -> mark#(U112(isNatIListKind(L),L,N))
p100: mark#(U102(X1,X2,X3)) -> mark#(X1)
p101: mark#(U101(X1,X2,X3)) -> active#(U101(mark(X1),X2,X3))
p102: active#(U11(tt(),V1)) -> mark#(U12(isNatIListKind(V1),V1))
p103: active#(U105(tt(),V2)) -> mark#(U106(isNatIList(V2)))
p104: active#(U104(tt(),V1,V2)) -> mark#(U105(isNat(V1),V2))
p105: active#(U103(tt(),V1,V2)) -> mark#(U104(isNatIListKind(V2),V1,V2))
p106: active#(U101(tt(),V1,V2)) -> mark#(U102(isNatKind(V1),V1,V2))

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41, r42, r43, r44, r45, r46, r47, r48, r49, r50, r51, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76, r77, r78, r79, r80, r81, r82, r83, r84, r85, r86, r87, r88, r89, r90, r91, r92, r93, r94, r95, r96, r97, r98, r99, r100, r101, r102, r103, r104, r105, r106, r107, r108, r109, r110, r111, r112, r113, r114, r115, r116, r117, r118, r119, r120, r121, r122, r123, r124, r125, r126, r127, r128, r129, r130, r131, r132, r133, r134, r135, r136, r137, r138, r139, r140, r141, r142, r143, r144, r145, r146, r147, r148, r149, r150, r151, r152, r153, r154, r155, r156, r157, r158, r159, r160, r161, r162, r163, r164, r165, r166, r167, r168, r169, r170, r171, r172, r173, r174, r175, r176, r177, r178, r179, r180, r181, r182, r183, r184, r185, r186, r187, r188, r189, r190, r191, r192, r193, r194, r195, r196, r197, r198, r199, r200, r201, r202, r203, r204, r205, r206, r207, r208, r209, r210, r211, r212, r213, r214, r215, r216, r217, r218, r219, r220, r221, r222, r223, r224, r225, r226, r227, r228, r229, r230, r231, r232, r233, r234, r235, r236, r237, r238, r239, r240, r241, r242, r243, r244, r245, r246, r247, r248, r249, r250, r251, r252, r253, r254, r255, r256, r257, r258, r259, r260, r261, r262, r263, r264, r265, r266, r267, r268, r269, r270, r271, r272, r273, r274, r275, r276, r277, r278, r279, r280, r281, r282, r283, r284, r285, r286, r287, r288, r289, r290, r291, r292, r293, r294, r295, r296, r297, r298, r299, r300, r301, r302, r303, r304, r305, r306, r307, r308, r309, r310, r311, r312, r313, r314, r315, r316, r317, r318, r319, r320, r321, r322, r323, r324, r325, r326, r327, r328, r329, r330, r331, r332, r333, r334, r335, r336, r337, r338, r339, r340, r341, r342, r343, r344, r345, r346, r347, r348, r349, r350, r351, r352, r353, r354, r355, r356, r357, r358

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    mark#_A(x1) = x1 + 2
    U96_A(x1) = max{13, x1}
    U95_A(x1,x2) = max{x1, x2 + 19}
    active#_A(x1) = x1 + 2
    mark_A(x1) = x1
    take_A(x1,x2) = max{x1 + 142, x2 + 184}
    s_A(x1) = max{5, x1}
    cons_A(x1,x2) = max{x1 + 50, x2}
    U131_A(x1,x2,x3,x4) = max{x1 + 141, x2 + 184, x3 + 142, x4 + 106}
    isNatIList_A(x1) = x1 + 42
    U94_A(x1,x2,x3) = max{x1 + 11, x2 + 45, x3 + 19}
    |0|_A = 4
    U121_A(x1,x2) = max{x1 + 8, x2 + 184}
    U93_A(x1,x2,x3) = max{x1 + 2, x2 + 45, x3 + 19}
    length_A(x1) = x1 + 40
    U111_A(x1,x2,x3) = max{x1 + 20, x2 + 40, x3 + 90}
    isNatList_A(x1) = x1 + 19
    U92_A(x1,x2,x3) = max{x1, x2 + 45, x3 + 19}
    U101_A(x1,x2,x3) = max{x1 + 77, x2 + 89, x3 + 203}
    isNatKind_A(x1) = x1 + 44
    U91_A(x1,x2,x3) = max{x1, x2 + 46, x3 + 19}
    U81_A(x1) = max{14, x1}
    U71_A(x1) = max{84, x1}
    U62_A(x1) = max{5, x1}
    U61_A(x1,x2) = max{x1 + 7, x2 + 7}
    U52_A(x1) = max{5, x1}
    U51_A(x1,x2) = max{x1, x2 + 7}
    isNatIListKind_A(x1) = x1 + 7
    U46_A(x1) = max{42, x1}
    U45_A(x1,x2) = max{x1, x2 + 42}
    U44_A(x1,x2,x3) = max{x1 + 35, x2 + 45, x3 + 42}
    U41_A(x1,x2,x3) = max{x1 + 47, x2 + 46, x3 + 42}
    U43_A(x1,x2,x3) = max{x1 + 7, x2 + 45, x3 + 42}
    isNat_A(x1) = x1 + 45
    U21_A(x1,x2) = max{x1, x2 + 45}
    U42_A(x1,x2,x3) = max{x1, x2 + 45, x3 + 42}
    U11_A(x1,x2) = max{x1 + 28, x2 + 29}
    tt_A = 14
    U33_A(x1) = max{19, x1}
    U32_A(x1,x2) = max{x1, x2 + 19}
    U23_A(x1) = max{14, x1}
    U22_A(x1,x2) = max{x1, x2 + 45}
    U136_A(x1,x2,x3,x4) = max{x1 + 52, x2 + 184, x3 + 142, x4 + 51}
    U135_A(x1,x2,x3,x4) = max{x1 + 51, x2 + 184, x3 + 142, x4 + 96}
    U134_A(x1,x2,x3,x4) = max{x1 + 49, x2 + 184, x3 + 142, x4 + 96}
    U133_A(x1,x2,x3,x4) = max{x1 + 97, x2 + 184, x3 + 142, x4 + 96}
    U132_A(x1,x2,x3,x4) = max{x1 + 98, x2 + 184, x3 + 142, x4 + 97}
    U122_A(x1) = max{184, x1}
    U13_A(x1) = max{29, x1}
    U114_A(x1,x2) = max{x1 + 24, x2 + 40}
    U113_A(x1,x2,x3) = max{x1 + 8, x2 + 40, x3 + 68}
    U112_A(x1,x2,x3) = max{x1 + 27, x2 + 40, x3 + 90}
    U12_A(x1,x2) = max{x1, x2 + 29}
    U106_A(x1) = max{40, x1}
    U105_A(x1,x2) = max{x1 + 43, x2 + 42}
    U104_A(x1,x2,x3) = max{x1 + 82, x2 + 88, x3 + 83}
    U103_A(x1,x2,x3) = max{x1, x2 + 89, x3 + 89}
    U102_A(x1,x2,x3) = max{x1, x2 + 89, x3 + 203}
    active_A(x1) = max{4, x1}
    zeros_A = 54
    nil_A = 7
    U31_A(x1,x2) = x2 + 19

The next rules are strictly ordered:

  p34

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(U96(X)) -> mark#(X)
p2: mark#(U95(X1,X2)) -> mark#(X1)
p3: mark#(U95(X1,X2)) -> active#(U95(mark(X1),X2))
p4: active#(take(s(M),cons(N,IL))) -> mark#(U131(isNatIList(IL),IL,M,N))
p5: mark#(U94(X1,X2,X3)) -> active#(U94(mark(X1),X2,X3))
p6: active#(take(|0|(),IL)) -> mark#(U121(isNatIList(IL),IL))
p7: mark#(U93(X1,X2,X3)) -> active#(U93(mark(X1),X2,X3))
p8: active#(length(cons(N,L))) -> mark#(U111(isNatList(L),L,N))
p9: mark#(U92(X1,X2,X3)) -> mark#(X1)
p10: mark#(U92(X1,X2,X3)) -> active#(U92(mark(X1),X2,X3))
p11: active#(isNatList(take(V1,V2))) -> mark#(U101(isNatKind(V1),V1,V2))
p12: mark#(U91(X1,X2,X3)) -> mark#(X1)
p13: mark#(U91(X1,X2,X3)) -> active#(U91(mark(X1),X2,X3))
p14: active#(isNatList(cons(V1,V2))) -> mark#(U91(isNatKind(V1),V1,V2))
p15: mark#(U81(X)) -> mark#(X)
p16: mark#(U71(X)) -> mark#(X)
p17: mark#(U62(X)) -> mark#(X)
p18: mark#(U61(X1,X2)) -> active#(U61(mark(X1),X2))
p19: active#(isNatKind(s(V1))) -> mark#(U81(isNatKind(V1)))
p20: mark#(U52(X)) -> mark#(X)
p21: mark#(U51(X1,X2)) -> mark#(X1)
p22: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p23: active#(isNatKind(length(V1))) -> mark#(U71(isNatIListKind(V1)))
p24: mark#(U46(X)) -> mark#(X)
p25: mark#(U45(X1,X2)) -> mark#(X1)
p26: mark#(U45(X1,X2)) -> active#(U45(mark(X1),X2))
p27: active#(isNatIListKind(cons(V1,V2))) -> mark#(U51(isNatKind(V1),V2))
p28: mark#(U44(X1,X2,X3)) -> active#(U44(mark(X1),X2,X3))
p29: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNatKind(V1),V1,V2))
p30: mark#(U43(X1,X2,X3)) -> active#(U43(mark(X1),X2,X3))
p31: active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1))
p32: mark#(U42(X1,X2,X3)) -> mark#(X1)
p33: mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3))
p34: mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3))
p35: active#(U95(tt(),V2)) -> mark#(U96(isNatList(V2)))
p36: mark#(U33(X)) -> mark#(X)
p37: mark#(U32(X1,X2)) -> mark#(X1)
p38: mark#(U32(X1,X2)) -> active#(U32(mark(X1),X2))
p39: active#(U94(tt(),V1,V2)) -> mark#(U95(isNat(V1),V2))
p40: mark#(U23(X)) -> mark#(X)
p41: mark#(U22(X1,X2)) -> mark#(X1)
p42: mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2))
p43: active#(U93(tt(),V1,V2)) -> mark#(U94(isNatIListKind(V2),V1,V2))
p44: mark#(U21(X1,X2)) -> mark#(X1)
p45: mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2))
p46: active#(U92(tt(),V1,V2)) -> mark#(U93(isNatIListKind(V2),V1,V2))
p47: mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2)))
p48: active#(U91(tt(),V1,V2)) -> mark#(U92(isNatKind(V1),V1,V2))
p49: mark#(U136(X1,X2,X3,X4)) -> active#(U136(mark(X1),X2,X3,X4))
p50: active#(U61(tt(),V2)) -> mark#(U62(isNatIListKind(V2)))
p51: mark#(U135(X1,X2,X3,X4)) -> active#(U135(mark(X1),X2,X3,X4))
p52: active#(U51(tt(),V2)) -> mark#(U52(isNatIListKind(V2)))
p53: mark#(U134(X1,X2,X3,X4)) -> active#(U134(mark(X1),X2,X3,X4))
p54: active#(U45(tt(),V2)) -> mark#(U46(isNatIList(V2)))
p55: mark#(U133(X1,X2,X3,X4)) -> active#(U133(mark(X1),X2,X3,X4))
p56: active#(U44(tt(),V1,V2)) -> mark#(U45(isNat(V1),V2))
p57: mark#(U132(X1,X2,X3,X4)) -> active#(U132(mark(X1),X2,X3,X4))
p58: active#(U43(tt(),V1,V2)) -> mark#(U44(isNatIListKind(V2),V1,V2))
p59: mark#(U131(X1,X2,X3,X4)) -> active#(U131(mark(X1),X2,X3,X4))
p60: active#(U42(tt(),V1,V2)) -> mark#(U43(isNatIListKind(V2),V1,V2))
p61: mark#(U122(X)) -> mark#(X)
p62: mark#(U121(X1,X2)) -> active#(U121(mark(X1),X2))
p63: active#(U41(tt(),V1,V2)) -> mark#(U42(isNatKind(V1),V1,V2))
p64: mark#(isNatList(X)) -> active#(isNatList(X))
p65: active#(U32(tt(),V)) -> mark#(U33(isNatList(V)))
p66: mark#(U13(X)) -> mark#(X)
p67: mark#(length(X)) -> active#(length(mark(X)))
p68: active#(U22(tt(),V1)) -> mark#(U23(isNat(V1)))
p69: mark#(s(X)) -> mark#(X)
p70: mark#(U114(X1,X2)) -> active#(U114(mark(X1),X2))
p71: active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1))
p72: mark#(U113(X1,X2,X3)) -> active#(U113(mark(X1),X2,X3))
p73: active#(U136(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL)))
p74: mark#(U112(X1,X2,X3)) -> active#(U112(mark(X1),X2,X3))
p75: active#(U135(tt(),IL,M,N)) -> mark#(U136(isNatKind(N),IL,M,N))
p76: mark#(U111(X1,X2,X3)) -> active#(U111(mark(X1),X2,X3))
p77: active#(U134(tt(),IL,M,N)) -> mark#(U135(isNat(N),IL,M,N))
p78: mark#(U12(X1,X2)) -> mark#(X1)
p79: mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2))
p80: active#(U133(tt(),IL,M,N)) -> mark#(U134(isNatKind(M),IL,M,N))
p81: mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2))
p82: active#(U132(tt(),IL,M,N)) -> mark#(U133(isNat(M),IL,M,N))
p83: mark#(isNatIList(X)) -> active#(isNatIList(X))
p84: active#(U131(tt(),IL,M,N)) -> mark#(U132(isNatIListKind(IL),IL,M,N))
p85: mark#(U106(X)) -> mark#(X)
p86: mark#(isNat(X)) -> active#(isNat(X))
p87: active#(U121(tt(),IL)) -> mark#(U122(isNatIListKind(IL)))
p88: mark#(U105(X1,X2)) -> active#(U105(mark(X1),X2))
p89: active#(U12(tt(),V1)) -> mark#(U13(isNatList(V1)))
p90: mark#(U104(X1,X2,X3)) -> active#(U104(mark(X1),X2,X3))
p91: active#(U114(tt(),L)) -> mark#(s(length(L)))
p92: mark#(isNatIListKind(X)) -> active#(isNatIListKind(X))
p93: active#(U113(tt(),L,N)) -> mark#(U114(isNatKind(N),L))
p94: mark#(U103(X1,X2,X3)) -> mark#(X1)
p95: mark#(U103(X1,X2,X3)) -> active#(U103(mark(X1),X2,X3))
p96: active#(U112(tt(),L,N)) -> mark#(U113(isNat(N),L,N))
p97: mark#(isNatKind(X)) -> active#(isNatKind(X))
p98: active#(U111(tt(),L,N)) -> mark#(U112(isNatIListKind(L),L,N))
p99: mark#(U102(X1,X2,X3)) -> mark#(X1)
p100: mark#(U101(X1,X2,X3)) -> active#(U101(mark(X1),X2,X3))
p101: active#(U11(tt(),V1)) -> mark#(U12(isNatIListKind(V1),V1))
p102: active#(U105(tt(),V2)) -> mark#(U106(isNatIList(V2)))
p103: active#(U104(tt(),V1,V2)) -> mark#(U105(isNat(V1),V2))
p104: active#(U103(tt(),V1,V2)) -> mark#(U104(isNatIListKind(V2),V1,V2))
p105: active#(U101(tt(),V1,V2)) -> mark#(U102(isNatKind(V1),V1,V2))

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19, p20, p21, p22, p23, p24, p25, p26, p27, p28, p29, p30, p31, p32, p33, p34, p35, p36, p37, p38, p39, p40, p41, p42, p43, p44, p45, p46, p47, p48, p49, p50, p51, p52, p53, p54, p55, p56, p57, p58, p59, p60, p61, p62, p63, p64, p65, p66, p67, p68, p69, p70, p71, p72, p73, p74, p75, p76, p77, p78, p79, p80, p81, p82, p83, p84, p85, p86, p87, p88, p89, p90, p91, p92, p93, p94, p95, p96, p97, p98, p99, p100, p101, p102, p103, p104, p105}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(U96(X)) -> mark#(X)
p2: mark#(U101(X1,X2,X3)) -> active#(U101(mark(X1),X2,X3))
p3: active#(U101(tt(),V1,V2)) -> mark#(U102(isNatKind(V1),V1,V2))
p4: mark#(U102(X1,X2,X3)) -> mark#(X1)
p5: mark#(isNatKind(X)) -> active#(isNatKind(X))
p6: active#(U103(tt(),V1,V2)) -> mark#(U104(isNatIListKind(V2),V1,V2))
p7: mark#(U103(X1,X2,X3)) -> active#(U103(mark(X1),X2,X3))
p8: active#(U104(tt(),V1,V2)) -> mark#(U105(isNat(V1),V2))
p9: mark#(U103(X1,X2,X3)) -> mark#(X1)
p10: mark#(isNatIListKind(X)) -> active#(isNatIListKind(X))
p11: active#(U105(tt(),V2)) -> mark#(U106(isNatIList(V2)))
p12: mark#(U104(X1,X2,X3)) -> active#(U104(mark(X1),X2,X3))
p13: active#(U11(tt(),V1)) -> mark#(U12(isNatIListKind(V1),V1))
p14: mark#(U105(X1,X2)) -> active#(U105(mark(X1),X2))
p15: active#(U111(tt(),L,N)) -> mark#(U112(isNatIListKind(L),L,N))
p16: mark#(isNat(X)) -> active#(isNat(X))
p17: active#(U112(tt(),L,N)) -> mark#(U113(isNat(N),L,N))
p18: mark#(U106(X)) -> mark#(X)
p19: mark#(isNatIList(X)) -> active#(isNatIList(X))
p20: active#(U113(tt(),L,N)) -> mark#(U114(isNatKind(N),L))
p21: mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2))
p22: active#(U114(tt(),L)) -> mark#(s(length(L)))
p23: mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2))
p24: active#(U12(tt(),V1)) -> mark#(U13(isNatList(V1)))
p25: mark#(U12(X1,X2)) -> mark#(X1)
p26: mark#(U111(X1,X2,X3)) -> active#(U111(mark(X1),X2,X3))
p27: active#(U121(tt(),IL)) -> mark#(U122(isNatIListKind(IL)))
p28: mark#(U112(X1,X2,X3)) -> active#(U112(mark(X1),X2,X3))
p29: active#(U131(tt(),IL,M,N)) -> mark#(U132(isNatIListKind(IL),IL,M,N))
p30: mark#(U113(X1,X2,X3)) -> active#(U113(mark(X1),X2,X3))
p31: active#(U132(tt(),IL,M,N)) -> mark#(U133(isNat(M),IL,M,N))
p32: mark#(U114(X1,X2)) -> active#(U114(mark(X1),X2))
p33: active#(U133(tt(),IL,M,N)) -> mark#(U134(isNatKind(M),IL,M,N))
p34: mark#(s(X)) -> mark#(X)
p35: mark#(length(X)) -> active#(length(mark(X)))
p36: active#(U134(tt(),IL,M,N)) -> mark#(U135(isNat(N),IL,M,N))
p37: mark#(U13(X)) -> mark#(X)
p38: mark#(isNatList(X)) -> active#(isNatList(X))
p39: active#(U135(tt(),IL,M,N)) -> mark#(U136(isNatKind(N),IL,M,N))
p40: mark#(U121(X1,X2)) -> active#(U121(mark(X1),X2))
p41: active#(U136(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL)))
p42: mark#(U122(X)) -> mark#(X)
p43: mark#(U131(X1,X2,X3,X4)) -> active#(U131(mark(X1),X2,X3,X4))
p44: active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1))
p45: mark#(U132(X1,X2,X3,X4)) -> active#(U132(mark(X1),X2,X3,X4))
p46: active#(U22(tt(),V1)) -> mark#(U23(isNat(V1)))
p47: mark#(U133(X1,X2,X3,X4)) -> active#(U133(mark(X1),X2,X3,X4))
p48: active#(U32(tt(),V)) -> mark#(U33(isNatList(V)))
p49: mark#(U134(X1,X2,X3,X4)) -> active#(U134(mark(X1),X2,X3,X4))
p50: active#(U41(tt(),V1,V2)) -> mark#(U42(isNatKind(V1),V1,V2))
p51: mark#(U135(X1,X2,X3,X4)) -> active#(U135(mark(X1),X2,X3,X4))
p52: active#(U42(tt(),V1,V2)) -> mark#(U43(isNatIListKind(V2),V1,V2))
p53: mark#(U136(X1,X2,X3,X4)) -> active#(U136(mark(X1),X2,X3,X4))
p54: active#(U43(tt(),V1,V2)) -> mark#(U44(isNatIListKind(V2),V1,V2))
p55: mark#(take(X1,X2)) -> active#(take(mark(X1),mark(X2)))
p56: active#(U44(tt(),V1,V2)) -> mark#(U45(isNat(V1),V2))
p57: mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2))
p58: active#(U45(tt(),V2)) -> mark#(U46(isNatIList(V2)))
p59: mark#(U21(X1,X2)) -> mark#(X1)
p60: mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2))
p61: active#(U51(tt(),V2)) -> mark#(U52(isNatIListKind(V2)))
p62: mark#(U22(X1,X2)) -> mark#(X1)
p63: mark#(U23(X)) -> mark#(X)
p64: mark#(U32(X1,X2)) -> active#(U32(mark(X1),X2))
p65: active#(U61(tt(),V2)) -> mark#(U62(isNatIListKind(V2)))
p66: mark#(U32(X1,X2)) -> mark#(X1)
p67: mark#(U33(X)) -> mark#(X)
p68: mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3))
p69: active#(U91(tt(),V1,V2)) -> mark#(U92(isNatKind(V1),V1,V2))
p70: mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3))
p71: active#(U92(tt(),V1,V2)) -> mark#(U93(isNatIListKind(V2),V1,V2))
p72: mark#(U42(X1,X2,X3)) -> mark#(X1)
p73: mark#(U43(X1,X2,X3)) -> active#(U43(mark(X1),X2,X3))
p74: active#(U93(tt(),V1,V2)) -> mark#(U94(isNatIListKind(V2),V1,V2))
p75: mark#(U44(X1,X2,X3)) -> active#(U44(mark(X1),X2,X3))
p76: active#(U94(tt(),V1,V2)) -> mark#(U95(isNat(V1),V2))
p77: mark#(U45(X1,X2)) -> active#(U45(mark(X1),X2))
p78: active#(U95(tt(),V2)) -> mark#(U96(isNatList(V2)))
p79: mark#(U45(X1,X2)) -> mark#(X1)
p80: mark#(U46(X)) -> mark#(X)
p81: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p82: active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1))
p83: mark#(U51(X1,X2)) -> mark#(X1)
p84: mark#(U52(X)) -> mark#(X)
p85: mark#(U61(X1,X2)) -> active#(U61(mark(X1),X2))
p86: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNatKind(V1),V1,V2))
p87: mark#(U62(X)) -> mark#(X)
p88: mark#(U71(X)) -> mark#(X)
p89: mark#(U81(X)) -> mark#(X)
p90: mark#(U91(X1,X2,X3)) -> active#(U91(mark(X1),X2,X3))
p91: active#(isNatIListKind(cons(V1,V2))) -> mark#(U51(isNatKind(V1),V2))
p92: mark#(U91(X1,X2,X3)) -> mark#(X1)
p93: mark#(U92(X1,X2,X3)) -> active#(U92(mark(X1),X2,X3))
p94: active#(isNatKind(length(V1))) -> mark#(U71(isNatIListKind(V1)))
p95: mark#(U92(X1,X2,X3)) -> mark#(X1)
p96: mark#(U93(X1,X2,X3)) -> active#(U93(mark(X1),X2,X3))
p97: active#(isNatKind(s(V1))) -> mark#(U81(isNatKind(V1)))
p98: mark#(U94(X1,X2,X3)) -> active#(U94(mark(X1),X2,X3))
p99: active#(isNatList(cons(V1,V2))) -> mark#(U91(isNatKind(V1),V1,V2))
p100: mark#(U95(X1,X2)) -> active#(U95(mark(X1),X2))
p101: active#(isNatList(take(V1,V2))) -> mark#(U101(isNatKind(V1),V1,V2))
p102: mark#(U95(X1,X2)) -> mark#(X1)
p103: active#(length(cons(N,L))) -> mark#(U111(isNatList(L),L,N))
p104: active#(take(|0|(),IL)) -> mark#(U121(isNatIList(IL),IL))
p105: active#(take(s(M),cons(N,IL))) -> mark#(U131(isNatIList(IL),IL,M,N))

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41, r42, r43, r44, r45, r46, r47, r48, r49, r50, r51, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76, r77, r78, r79, r80, r81, r82, r83, r84, r85, r86, r87, r88, r89, r90, r91, r92, r93, r94, r95, r96, r97, r98, r99, r100, r101, r102, r103, r104, r105, r106, r107, r108, r109, r110, r111, r112, r113, r114, r115, r116, r117, r118, r119, r120, r121, r122, r123, r124, r125, r126, r127, r128, r129, r130, r131, r132, r133, r134, r135, r136, r137, r138, r139, r140, r141, r142, r143, r144, r145, r146, r147, r148, r149, r150, r151, r152, r153, r154, r155, r156, r157, r158, r159, r160, r161, r162, r163, r164, r165, r166, r167, r168, r169, r170, r171, r172, r173, r174, r175, r176, r177, r178, r179, r180, r181, r182, r183, r184, r185, r186, r187, r188, r189, r190, r191, r192, r193, r194, r195, r196, r197, r198, r199, r200, r201, r202, r203, r204, r205, r206, r207, r208, r209, r210, r211, r212, r213, r214, r215, r216, r217, r218, r219, r220, r221, r222, r223, r224, r225, r226, r227, r228, r229, r230, r231, r232, r233, r234, r235, r236, r237, r238, r239, r240, r241, r242, r243, r244, r245, r246, r247, r248, r249, r250, r251, r252, r253, r254, r255, r256, r257, r258, r259, r260, r261, r262, r263, r264, r265, r266, r267, r268, r269, r270, r271, r272, r273, r274, r275, r276, r277, r278, r279, r280, r281, r282, r283, r284, r285, r286, r287, r288, r289, r290, r291, r292, r293, r294, r295, r296, r297, r298, r299, r300, r301, r302, r303, r304, r305, r306, r307, r308, r309, r310, r311, r312, r313, r314, r315, r316, r317, r318, r319, r320, r321, r322, r323, r324, r325, r326, r327, r328, r329, r330, r331, r332, r333, r334, r335, r336, r337, r338, r339, r340, r341, r342, r343, r344, r345, r346, r347, r348, r349, r350, r351, r352, r353, r354, r355, r356, r357, r358

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    mark#_A(x1) = max{0, x1 - 49}
    U96_A(x1) = max{41, x1}
    U101_A(x1,x2,x3) = 41
    active#_A(x1) = max{0, x1 - 131}
    mark_A(x1) = max{36, x1 + 31}
    tt_A = 41
    U102_A(x1,x2,x3) = max{41, x1}
    isNatKind_A(x1) = 41
    U103_A(x1,x2,x3) = max{41, x1}
    U104_A(x1,x2,x3) = 41
    isNatIListKind_A(x1) = 41
    U105_A(x1,x2) = max{41, x1 - 31}
    isNat_A(x1) = 41
    U106_A(x1) = max{41, x1}
    isNatIList_A(x1) = 41
    U11_A(x1,x2) = max{41, x1 - 30}
    U12_A(x1,x2) = max{41, x1}
    U111_A(x1,x2,x3) = 41
    U112_A(x1,x2,x3) = max{41, x1 - 6}
    U113_A(x1,x2,x3) = max{41, x1 - 89}
    U114_A(x1,x2) = max{41, x1 - 42}
    s_A(x1) = max{41, x1}
    length_A(x1) = 41
    U13_A(x1) = max{41, x1}
    isNatList_A(x1) = 41
    U121_A(x1,x2) = max{444, x1 + 406}
    U122_A(x1) = max{365, x1 + 322}
    U131_A(x1,x2,x3,x4) = max{48, x1 + 1, x3 + 6}
    U132_A(x1,x2,x3,x4) = max{41, x1 - 30, x3 - 77}
    U133_A(x1,x2,x3,x4) = max{41, x1 - 83, x3 - 160}
    U134_A(x1,x2,x3,x4) = max{41, x1 - 31, x3 - 243}
    U135_A(x1,x2,x3,x4) = max{41, x3 - 325}
    U136_A(x1,x2,x3,x4) = max{41, x1 - 8, x3 - 407}
    cons_A(x1,x2) = 41
    take_A(x1,x2) = max{529, x1 + 88, x2 + 35}
    U21_A(x1,x2) = max{41, x1}
    U22_A(x1,x2) = max{41, x1}
    U23_A(x1) = max{41, x1}
    U32_A(x1,x2) = max{41, x1}
    U33_A(x1) = max{41, x1}
    U41_A(x1,x2,x3) = 41
    U42_A(x1,x2,x3) = max{41, x1}
    U43_A(x1,x2,x3) = max{41, x1 - 10}
    U44_A(x1,x2,x3) = max{41, x1 - 40}
    U45_A(x1,x2) = max{41, x1}
    U46_A(x1) = max{41, x1}
    U51_A(x1,x2) = max{41, x1}
    U52_A(x1) = max{41, x1}
    U61_A(x1,x2) = 41
    U62_A(x1) = max{41, x1}
    U91_A(x1,x2,x3) = max{41, x1}
    U92_A(x1,x2,x3) = max{41, x1}
    U93_A(x1,x2,x3) = 41
    U94_A(x1,x2,x3) = 41
    U95_A(x1,x2) = max{41, x1}
    U71_A(x1) = max{41, x1}
    U81_A(x1) = max{41, x1}
    |0|_A = 41
    active_A(x1) = max{72, x1}
    zeros_A = 619
    nil_A = 333
    U31_A(x1,x2) = max{41, x1}

The next rules are strictly ordered:

  p40, p42, p55

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(U96(X)) -> mark#(X)
p2: mark#(U101(X1,X2,X3)) -> active#(U101(mark(X1),X2,X3))
p3: active#(U101(tt(),V1,V2)) -> mark#(U102(isNatKind(V1),V1,V2))
p4: mark#(U102(X1,X2,X3)) -> mark#(X1)
p5: mark#(isNatKind(X)) -> active#(isNatKind(X))
p6: active#(U103(tt(),V1,V2)) -> mark#(U104(isNatIListKind(V2),V1,V2))
p7: mark#(U103(X1,X2,X3)) -> active#(U103(mark(X1),X2,X3))
p8: active#(U104(tt(),V1,V2)) -> mark#(U105(isNat(V1),V2))
p9: mark#(U103(X1,X2,X3)) -> mark#(X1)
p10: mark#(isNatIListKind(X)) -> active#(isNatIListKind(X))
p11: active#(U105(tt(),V2)) -> mark#(U106(isNatIList(V2)))
p12: mark#(U104(X1,X2,X3)) -> active#(U104(mark(X1),X2,X3))
p13: active#(U11(tt(),V1)) -> mark#(U12(isNatIListKind(V1),V1))
p14: mark#(U105(X1,X2)) -> active#(U105(mark(X1),X2))
p15: active#(U111(tt(),L,N)) -> mark#(U112(isNatIListKind(L),L,N))
p16: mark#(isNat(X)) -> active#(isNat(X))
p17: active#(U112(tt(),L,N)) -> mark#(U113(isNat(N),L,N))
p18: mark#(U106(X)) -> mark#(X)
p19: mark#(isNatIList(X)) -> active#(isNatIList(X))
p20: active#(U113(tt(),L,N)) -> mark#(U114(isNatKind(N),L))
p21: mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2))
p22: active#(U114(tt(),L)) -> mark#(s(length(L)))
p23: mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2))
p24: active#(U12(tt(),V1)) -> mark#(U13(isNatList(V1)))
p25: mark#(U12(X1,X2)) -> mark#(X1)
p26: mark#(U111(X1,X2,X3)) -> active#(U111(mark(X1),X2,X3))
p27: active#(U121(tt(),IL)) -> mark#(U122(isNatIListKind(IL)))
p28: mark#(U112(X1,X2,X3)) -> active#(U112(mark(X1),X2,X3))
p29: active#(U131(tt(),IL,M,N)) -> mark#(U132(isNatIListKind(IL),IL,M,N))
p30: mark#(U113(X1,X2,X3)) -> active#(U113(mark(X1),X2,X3))
p31: active#(U132(tt(),IL,M,N)) -> mark#(U133(isNat(M),IL,M,N))
p32: mark#(U114(X1,X2)) -> active#(U114(mark(X1),X2))
p33: active#(U133(tt(),IL,M,N)) -> mark#(U134(isNatKind(M),IL,M,N))
p34: mark#(s(X)) -> mark#(X)
p35: mark#(length(X)) -> active#(length(mark(X)))
p36: active#(U134(tt(),IL,M,N)) -> mark#(U135(isNat(N),IL,M,N))
p37: mark#(U13(X)) -> mark#(X)
p38: mark#(isNatList(X)) -> active#(isNatList(X))
p39: active#(U135(tt(),IL,M,N)) -> mark#(U136(isNatKind(N),IL,M,N))
p40: active#(U136(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL)))
p41: mark#(U131(X1,X2,X3,X4)) -> active#(U131(mark(X1),X2,X3,X4))
p42: active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1))
p43: mark#(U132(X1,X2,X3,X4)) -> active#(U132(mark(X1),X2,X3,X4))
p44: active#(U22(tt(),V1)) -> mark#(U23(isNat(V1)))
p45: mark#(U133(X1,X2,X3,X4)) -> active#(U133(mark(X1),X2,X3,X4))
p46: active#(U32(tt(),V)) -> mark#(U33(isNatList(V)))
p47: mark#(U134(X1,X2,X3,X4)) -> active#(U134(mark(X1),X2,X3,X4))
p48: active#(U41(tt(),V1,V2)) -> mark#(U42(isNatKind(V1),V1,V2))
p49: mark#(U135(X1,X2,X3,X4)) -> active#(U135(mark(X1),X2,X3,X4))
p50: active#(U42(tt(),V1,V2)) -> mark#(U43(isNatIListKind(V2),V1,V2))
p51: mark#(U136(X1,X2,X3,X4)) -> active#(U136(mark(X1),X2,X3,X4))
p52: active#(U43(tt(),V1,V2)) -> mark#(U44(isNatIListKind(V2),V1,V2))
p53: active#(U44(tt(),V1,V2)) -> mark#(U45(isNat(V1),V2))
p54: mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2))
p55: active#(U45(tt(),V2)) -> mark#(U46(isNatIList(V2)))
p56: mark#(U21(X1,X2)) -> mark#(X1)
p57: mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2))
p58: active#(U51(tt(),V2)) -> mark#(U52(isNatIListKind(V2)))
p59: mark#(U22(X1,X2)) -> mark#(X1)
p60: mark#(U23(X)) -> mark#(X)
p61: mark#(U32(X1,X2)) -> active#(U32(mark(X1),X2))
p62: active#(U61(tt(),V2)) -> mark#(U62(isNatIListKind(V2)))
p63: mark#(U32(X1,X2)) -> mark#(X1)
p64: mark#(U33(X)) -> mark#(X)
p65: mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3))
p66: active#(U91(tt(),V1,V2)) -> mark#(U92(isNatKind(V1),V1,V2))
p67: mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3))
p68: active#(U92(tt(),V1,V2)) -> mark#(U93(isNatIListKind(V2),V1,V2))
p69: mark#(U42(X1,X2,X3)) -> mark#(X1)
p70: mark#(U43(X1,X2,X3)) -> active#(U43(mark(X1),X2,X3))
p71: active#(U93(tt(),V1,V2)) -> mark#(U94(isNatIListKind(V2),V1,V2))
p72: mark#(U44(X1,X2,X3)) -> active#(U44(mark(X1),X2,X3))
p73: active#(U94(tt(),V1,V2)) -> mark#(U95(isNat(V1),V2))
p74: mark#(U45(X1,X2)) -> active#(U45(mark(X1),X2))
p75: active#(U95(tt(),V2)) -> mark#(U96(isNatList(V2)))
p76: mark#(U45(X1,X2)) -> mark#(X1)
p77: mark#(U46(X)) -> mark#(X)
p78: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p79: active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1))
p80: mark#(U51(X1,X2)) -> mark#(X1)
p81: mark#(U52(X)) -> mark#(X)
p82: mark#(U61(X1,X2)) -> active#(U61(mark(X1),X2))
p83: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNatKind(V1),V1,V2))
p84: mark#(U62(X)) -> mark#(X)
p85: mark#(U71(X)) -> mark#(X)
p86: mark#(U81(X)) -> mark#(X)
p87: mark#(U91(X1,X2,X3)) -> active#(U91(mark(X1),X2,X3))
p88: active#(isNatIListKind(cons(V1,V2))) -> mark#(U51(isNatKind(V1),V2))
p89: mark#(U91(X1,X2,X3)) -> mark#(X1)
p90: mark#(U92(X1,X2,X3)) -> active#(U92(mark(X1),X2,X3))
p91: active#(isNatKind(length(V1))) -> mark#(U71(isNatIListKind(V1)))
p92: mark#(U92(X1,X2,X3)) -> mark#(X1)
p93: mark#(U93(X1,X2,X3)) -> active#(U93(mark(X1),X2,X3))
p94: active#(isNatKind(s(V1))) -> mark#(U81(isNatKind(V1)))
p95: mark#(U94(X1,X2,X3)) -> active#(U94(mark(X1),X2,X3))
p96: active#(isNatList(cons(V1,V2))) -> mark#(U91(isNatKind(V1),V1,V2))
p97: mark#(U95(X1,X2)) -> active#(U95(mark(X1),X2))
p98: active#(isNatList(take(V1,V2))) -> mark#(U101(isNatKind(V1),V1,V2))
p99: mark#(U95(X1,X2)) -> mark#(X1)
p100: active#(length(cons(N,L))) -> mark#(U111(isNatList(L),L,N))
p101: active#(take(|0|(),IL)) -> mark#(U121(isNatIList(IL),IL))
p102: active#(take(s(M),cons(N,IL))) -> mark#(U131(isNatIList(IL),IL,M,N))

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19, p20, p21, p22, p23, p24, p25, p26, p27, p28, p29, p30, p31, p32, p33, p34, p35, p36, p37, p38, p39, p40, p41, p42, p43, p44, p45, p46, p47, p48, p49, p50, p51, p52, p53, p54, p55, p56, p57, p58, p59, p60, p61, p62, p63, p64, p65, p66, p67, p68, p69, p70, p71, p72, p73, p74, p75, p76, p77, p78, p79, p80, p81, p82, p83, p84, p85, p86, p87, p88, p89, p90, p91, p92, p93, p94, p95, p96, p97, p98, p99, p100, p101, p102}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(U96(X)) -> mark#(X)
p2: mark#(U95(X1,X2)) -> mark#(X1)
p3: mark#(U95(X1,X2)) -> active#(U95(mark(X1),X2))
p4: active#(take(s(M),cons(N,IL))) -> mark#(U131(isNatIList(IL),IL,M,N))
p5: mark#(U94(X1,X2,X3)) -> active#(U94(mark(X1),X2,X3))
p6: active#(take(|0|(),IL)) -> mark#(U121(isNatIList(IL),IL))
p7: mark#(U93(X1,X2,X3)) -> active#(U93(mark(X1),X2,X3))
p8: active#(length(cons(N,L))) -> mark#(U111(isNatList(L),L,N))
p9: mark#(U92(X1,X2,X3)) -> mark#(X1)
p10: mark#(U92(X1,X2,X3)) -> active#(U92(mark(X1),X2,X3))
p11: active#(isNatList(take(V1,V2))) -> mark#(U101(isNatKind(V1),V1,V2))
p12: mark#(U91(X1,X2,X3)) -> mark#(X1)
p13: mark#(U91(X1,X2,X3)) -> active#(U91(mark(X1),X2,X3))
p14: active#(isNatList(cons(V1,V2))) -> mark#(U91(isNatKind(V1),V1,V2))
p15: mark#(U81(X)) -> mark#(X)
p16: mark#(U71(X)) -> mark#(X)
p17: mark#(U62(X)) -> mark#(X)
p18: mark#(U61(X1,X2)) -> active#(U61(mark(X1),X2))
p19: active#(isNatKind(s(V1))) -> mark#(U81(isNatKind(V1)))
p20: mark#(U52(X)) -> mark#(X)
p21: mark#(U51(X1,X2)) -> mark#(X1)
p22: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p23: active#(isNatKind(length(V1))) -> mark#(U71(isNatIListKind(V1)))
p24: mark#(U46(X)) -> mark#(X)
p25: mark#(U45(X1,X2)) -> mark#(X1)
p26: mark#(U45(X1,X2)) -> active#(U45(mark(X1),X2))
p27: active#(isNatIListKind(cons(V1,V2))) -> mark#(U51(isNatKind(V1),V2))
p28: mark#(U44(X1,X2,X3)) -> active#(U44(mark(X1),X2,X3))
p29: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNatKind(V1),V1,V2))
p30: mark#(U43(X1,X2,X3)) -> active#(U43(mark(X1),X2,X3))
p31: active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1))
p32: mark#(U42(X1,X2,X3)) -> mark#(X1)
p33: mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3))
p34: active#(U95(tt(),V2)) -> mark#(U96(isNatList(V2)))
p35: mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3))
p36: active#(U94(tt(),V1,V2)) -> mark#(U95(isNat(V1),V2))
p37: mark#(U33(X)) -> mark#(X)
p38: mark#(U32(X1,X2)) -> mark#(X1)
p39: mark#(U32(X1,X2)) -> active#(U32(mark(X1),X2))
p40: active#(U93(tt(),V1,V2)) -> mark#(U94(isNatIListKind(V2),V1,V2))
p41: mark#(U23(X)) -> mark#(X)
p42: mark#(U22(X1,X2)) -> mark#(X1)
p43: mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2))
p44: active#(U92(tt(),V1,V2)) -> mark#(U93(isNatIListKind(V2),V1,V2))
p45: mark#(U21(X1,X2)) -> mark#(X1)
p46: mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2))
p47: active#(U91(tt(),V1,V2)) -> mark#(U92(isNatKind(V1),V1,V2))
p48: mark#(U136(X1,X2,X3,X4)) -> active#(U136(mark(X1),X2,X3,X4))
p49: active#(U61(tt(),V2)) -> mark#(U62(isNatIListKind(V2)))
p50: mark#(U135(X1,X2,X3,X4)) -> active#(U135(mark(X1),X2,X3,X4))
p51: active#(U51(tt(),V2)) -> mark#(U52(isNatIListKind(V2)))
p52: mark#(U134(X1,X2,X3,X4)) -> active#(U134(mark(X1),X2,X3,X4))
p53: active#(U45(tt(),V2)) -> mark#(U46(isNatIList(V2)))
p54: mark#(U133(X1,X2,X3,X4)) -> active#(U133(mark(X1),X2,X3,X4))
p55: active#(U44(tt(),V1,V2)) -> mark#(U45(isNat(V1),V2))
p56: mark#(U132(X1,X2,X3,X4)) -> active#(U132(mark(X1),X2,X3,X4))
p57: active#(U43(tt(),V1,V2)) -> mark#(U44(isNatIListKind(V2),V1,V2))
p58: mark#(U131(X1,X2,X3,X4)) -> active#(U131(mark(X1),X2,X3,X4))
p59: active#(U42(tt(),V1,V2)) -> mark#(U43(isNatIListKind(V2),V1,V2))
p60: mark#(isNatList(X)) -> active#(isNatList(X))
p61: active#(U41(tt(),V1,V2)) -> mark#(U42(isNatKind(V1),V1,V2))
p62: mark#(U13(X)) -> mark#(X)
p63: mark#(length(X)) -> active#(length(mark(X)))
p64: active#(U32(tt(),V)) -> mark#(U33(isNatList(V)))
p65: mark#(s(X)) -> mark#(X)
p66: mark#(U114(X1,X2)) -> active#(U114(mark(X1),X2))
p67: active#(U22(tt(),V1)) -> mark#(U23(isNat(V1)))
p68: mark#(U113(X1,X2,X3)) -> active#(U113(mark(X1),X2,X3))
p69: active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1))
p70: mark#(U112(X1,X2,X3)) -> active#(U112(mark(X1),X2,X3))
p71: active#(U136(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL)))
p72: mark#(U111(X1,X2,X3)) -> active#(U111(mark(X1),X2,X3))
p73: active#(U135(tt(),IL,M,N)) -> mark#(U136(isNatKind(N),IL,M,N))
p74: mark#(U12(X1,X2)) -> mark#(X1)
p75: mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2))
p76: active#(U134(tt(),IL,M,N)) -> mark#(U135(isNat(N),IL,M,N))
p77: mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2))
p78: active#(U133(tt(),IL,M,N)) -> mark#(U134(isNatKind(M),IL,M,N))
p79: mark#(isNatIList(X)) -> active#(isNatIList(X))
p80: active#(U132(tt(),IL,M,N)) -> mark#(U133(isNat(M),IL,M,N))
p81: mark#(U106(X)) -> mark#(X)
p82: mark#(isNat(X)) -> active#(isNat(X))
p83: active#(U131(tt(),IL,M,N)) -> mark#(U132(isNatIListKind(IL),IL,M,N))
p84: mark#(U105(X1,X2)) -> active#(U105(mark(X1),X2))
p85: active#(U121(tt(),IL)) -> mark#(U122(isNatIListKind(IL)))
p86: mark#(U104(X1,X2,X3)) -> active#(U104(mark(X1),X2,X3))
p87: active#(U12(tt(),V1)) -> mark#(U13(isNatList(V1)))
p88: mark#(isNatIListKind(X)) -> active#(isNatIListKind(X))
p89: active#(U114(tt(),L)) -> mark#(s(length(L)))
p90: mark#(U103(X1,X2,X3)) -> mark#(X1)
p91: mark#(U103(X1,X2,X3)) -> active#(U103(mark(X1),X2,X3))
p92: active#(U113(tt(),L,N)) -> mark#(U114(isNatKind(N),L))
p93: mark#(isNatKind(X)) -> active#(isNatKind(X))
p94: active#(U112(tt(),L,N)) -> mark#(U113(isNat(N),L,N))
p95: mark#(U102(X1,X2,X3)) -> mark#(X1)
p96: mark#(U101(X1,X2,X3)) -> active#(U101(mark(X1),X2,X3))
p97: active#(U111(tt(),L,N)) -> mark#(U112(isNatIListKind(L),L,N))
p98: active#(U11(tt(),V1)) -> mark#(U12(isNatIListKind(V1),V1))
p99: active#(U105(tt(),V2)) -> mark#(U106(isNatIList(V2)))
p100: active#(U104(tt(),V1,V2)) -> mark#(U105(isNat(V1),V2))
p101: active#(U103(tt(),V1,V2)) -> mark#(U104(isNatIListKind(V2),V1,V2))
p102: active#(U101(tt(),V1,V2)) -> mark#(U102(isNatKind(V1),V1,V2))

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41, r42, r43, r44, r45, r46, r47, r48, r49, r50, r51, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76, r77, r78, r79, r80, r81, r82, r83, r84, r85, r86, r87, r88, r89, r90, r91, r92, r93, r94, r95, r96, r97, r98, r99, r100, r101, r102, r103, r104, r105, r106, r107, r108, r109, r110, r111, r112, r113, r114, r115, r116, r117, r118, r119, r120, r121, r122, r123, r124, r125, r126, r127, r128, r129, r130, r131, r132, r133, r134, r135, r136, r137, r138, r139, r140, r141, r142, r143, r144, r145, r146, r147, r148, r149, r150, r151, r152, r153, r154, r155, r156, r157, r158, r159, r160, r161, r162, r163, r164, r165, r166, r167, r168, r169, r170, r171, r172, r173, r174, r175, r176, r177, r178, r179, r180, r181, r182, r183, r184, r185, r186, r187, r188, r189, r190, r191, r192, r193, r194, r195, r196, r197, r198, r199, r200, r201, r202, r203, r204, r205, r206, r207, r208, r209, r210, r211, r212, r213, r214, r215, r216, r217, r218, r219, r220, r221, r222, r223, r224, r225, r226, r227, r228, r229, r230, r231, r232, r233, r234, r235, r236, r237, r238, r239, r240, r241, r242, r243, r244, r245, r246, r247, r248, r249, r250, r251, r252, r253, r254, r255, r256, r257, r258, r259, r260, r261, r262, r263, r264, r265, r266, r267, r268, r269, r270, r271, r272, r273, r274, r275, r276, r277, r278, r279, r280, r281, r282, r283, r284, r285, r286, r287, r288, r289, r290, r291, r292, r293, r294, r295, r296, r297, r298, r299, r300, r301, r302, r303, r304, r305, r306, r307, r308, r309, r310, r311, r312, r313, r314, r315, r316, r317, r318, r319, r320, r321, r322, r323, r324, r325, r326, r327, r328, r329, r330, r331, r332, r333, r334, r335, r336, r337, r338, r339, r340, r341, r342, r343, r344, r345, r346, r347, r348, r349, r350, r351, r352, r353, r354, r355, r356, r357, r358

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    mark#_A(x1) = 0
    U96_A(x1) = max{564, x1 - 10}
    U95_A(x1,x2) = 628
    active#_A(x1) = max{0, x1 - 638}
    mark_A(x1) = max{497, x1}
    take_A(x1,x2) = max{703, x1 + 206, x2 + 114}
    s_A(x1) = max{638, x1 - 1}
    cons_A(x1,x2) = 602
    U131_A(x1,x2,x3,x4) = 638
    isNatIList_A(x1) = 629
    U94_A(x1,x2,x3) = 637
    |0|_A = 638
    U121_A(x1,x2) = max{711, x1 + 147}
    U93_A(x1,x2,x3) = 638
    length_A(x1) = 638
    U111_A(x1,x2,x3) = 638
    isNatList_A(x1) = 638
    U92_A(x1,x2,x3) = 638
    U101_A(x1,x2,x3) = 638
    isNatKind_A(x1) = 634
    U91_A(x1,x2,x3) = 638
    U81_A(x1) = max{628, x1 - 67}
    U71_A(x1) = max{628, x1 - 1}
    U62_A(x1) = 628
    U61_A(x1,x2) = 628
    U52_A(x1) = 628
    U51_A(x1,x2) = 628
    isNatIListKind_A(x1) = 630
    U46_A(x1) = 564
    U45_A(x1,x2) = 564
    U44_A(x1,x2,x3) = 628
    U41_A(x1,x2,x3) = 629
    U43_A(x1,x2,x3) = 628
    isNat_A(x1) = 601
    U21_A(x1,x2) = 589
    U42_A(x1,x2,x3) = 629
    tt_A = 564
    U33_A(x1) = max{564, x1 - 205}
    U32_A(x1,x2) = 626
    U23_A(x1) = 565
    U22_A(x1,x2) = 577
    U136_A(x1,x2,x3,x4) = 614
    U135_A(x1,x2,x3,x4) = 626
    U134_A(x1,x2,x3,x4) = 632
    U133_A(x1,x2,x3,x4) = 638
    U132_A(x1,x2,x3,x4) = 638
    U13_A(x1) = max{565, x1 - 61}
    U114_A(x1,x2) = 638
    U113_A(x1,x2,x3) = 638
    U112_A(x1,x2,x3) = 638
    U12_A(x1,x2) = 577
    U11_A(x1,x2) = 601
    U106_A(x1) = max{578, x1 - 64}
    U105_A(x1,x2) = 590
    U122_A(x1) = 565
    U104_A(x1,x2,x3) = 626
    U103_A(x1,x2,x3) = 638
    U102_A(x1,x2,x3) = max{638, x1 + 1}
    active_A(x1) = max{564, x1}
    zeros_A = 1020
    nil_A = 565
    U31_A(x1,x2) = max{626, x1 - 66}

The next rules are strictly ordered:

  p4, p6, p85

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(U96(X)) -> mark#(X)
p2: mark#(U95(X1,X2)) -> mark#(X1)
p3: mark#(U95(X1,X2)) -> active#(U95(mark(X1),X2))
p4: mark#(U94(X1,X2,X3)) -> active#(U94(mark(X1),X2,X3))
p5: mark#(U93(X1,X2,X3)) -> active#(U93(mark(X1),X2,X3))
p6: active#(length(cons(N,L))) -> mark#(U111(isNatList(L),L,N))
p7: mark#(U92(X1,X2,X3)) -> mark#(X1)
p8: mark#(U92(X1,X2,X3)) -> active#(U92(mark(X1),X2,X3))
p9: active#(isNatList(take(V1,V2))) -> mark#(U101(isNatKind(V1),V1,V2))
p10: mark#(U91(X1,X2,X3)) -> mark#(X1)
p11: mark#(U91(X1,X2,X3)) -> active#(U91(mark(X1),X2,X3))
p12: active#(isNatList(cons(V1,V2))) -> mark#(U91(isNatKind(V1),V1,V2))
p13: mark#(U81(X)) -> mark#(X)
p14: mark#(U71(X)) -> mark#(X)
p15: mark#(U62(X)) -> mark#(X)
p16: mark#(U61(X1,X2)) -> active#(U61(mark(X1),X2))
p17: active#(isNatKind(s(V1))) -> mark#(U81(isNatKind(V1)))
p18: mark#(U52(X)) -> mark#(X)
p19: mark#(U51(X1,X2)) -> mark#(X1)
p20: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p21: active#(isNatKind(length(V1))) -> mark#(U71(isNatIListKind(V1)))
p22: mark#(U46(X)) -> mark#(X)
p23: mark#(U45(X1,X2)) -> mark#(X1)
p24: mark#(U45(X1,X2)) -> active#(U45(mark(X1),X2))
p25: active#(isNatIListKind(cons(V1,V2))) -> mark#(U51(isNatKind(V1),V2))
p26: mark#(U44(X1,X2,X3)) -> active#(U44(mark(X1),X2,X3))
p27: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNatKind(V1),V1,V2))
p28: mark#(U43(X1,X2,X3)) -> active#(U43(mark(X1),X2,X3))
p29: active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1))
p30: mark#(U42(X1,X2,X3)) -> mark#(X1)
p31: mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3))
p32: active#(U95(tt(),V2)) -> mark#(U96(isNatList(V2)))
p33: mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3))
p34: active#(U94(tt(),V1,V2)) -> mark#(U95(isNat(V1),V2))
p35: mark#(U33(X)) -> mark#(X)
p36: mark#(U32(X1,X2)) -> mark#(X1)
p37: mark#(U32(X1,X2)) -> active#(U32(mark(X1),X2))
p38: active#(U93(tt(),V1,V2)) -> mark#(U94(isNatIListKind(V2),V1,V2))
p39: mark#(U23(X)) -> mark#(X)
p40: mark#(U22(X1,X2)) -> mark#(X1)
p41: mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2))
p42: active#(U92(tt(),V1,V2)) -> mark#(U93(isNatIListKind(V2),V1,V2))
p43: mark#(U21(X1,X2)) -> mark#(X1)
p44: mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2))
p45: active#(U91(tt(),V1,V2)) -> mark#(U92(isNatKind(V1),V1,V2))
p46: mark#(U136(X1,X2,X3,X4)) -> active#(U136(mark(X1),X2,X3,X4))
p47: active#(U61(tt(),V2)) -> mark#(U62(isNatIListKind(V2)))
p48: mark#(U135(X1,X2,X3,X4)) -> active#(U135(mark(X1),X2,X3,X4))
p49: active#(U51(tt(),V2)) -> mark#(U52(isNatIListKind(V2)))
p50: mark#(U134(X1,X2,X3,X4)) -> active#(U134(mark(X1),X2,X3,X4))
p51: active#(U45(tt(),V2)) -> mark#(U46(isNatIList(V2)))
p52: mark#(U133(X1,X2,X3,X4)) -> active#(U133(mark(X1),X2,X3,X4))
p53: active#(U44(tt(),V1,V2)) -> mark#(U45(isNat(V1),V2))
p54: mark#(U132(X1,X2,X3,X4)) -> active#(U132(mark(X1),X2,X3,X4))
p55: active#(U43(tt(),V1,V2)) -> mark#(U44(isNatIListKind(V2),V1,V2))
p56: mark#(U131(X1,X2,X3,X4)) -> active#(U131(mark(X1),X2,X3,X4))
p57: active#(U42(tt(),V1,V2)) -> mark#(U43(isNatIListKind(V2),V1,V2))
p58: mark#(isNatList(X)) -> active#(isNatList(X))
p59: active#(U41(tt(),V1,V2)) -> mark#(U42(isNatKind(V1),V1,V2))
p60: mark#(U13(X)) -> mark#(X)
p61: mark#(length(X)) -> active#(length(mark(X)))
p62: active#(U32(tt(),V)) -> mark#(U33(isNatList(V)))
p63: mark#(s(X)) -> mark#(X)
p64: mark#(U114(X1,X2)) -> active#(U114(mark(X1),X2))
p65: active#(U22(tt(),V1)) -> mark#(U23(isNat(V1)))
p66: mark#(U113(X1,X2,X3)) -> active#(U113(mark(X1),X2,X3))
p67: active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1))
p68: mark#(U112(X1,X2,X3)) -> active#(U112(mark(X1),X2,X3))
p69: active#(U136(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL)))
p70: mark#(U111(X1,X2,X3)) -> active#(U111(mark(X1),X2,X3))
p71: active#(U135(tt(),IL,M,N)) -> mark#(U136(isNatKind(N),IL,M,N))
p72: mark#(U12(X1,X2)) -> mark#(X1)
p73: mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2))
p74: active#(U134(tt(),IL,M,N)) -> mark#(U135(isNat(N),IL,M,N))
p75: mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2))
p76: active#(U133(tt(),IL,M,N)) -> mark#(U134(isNatKind(M),IL,M,N))
p77: mark#(isNatIList(X)) -> active#(isNatIList(X))
p78: active#(U132(tt(),IL,M,N)) -> mark#(U133(isNat(M),IL,M,N))
p79: mark#(U106(X)) -> mark#(X)
p80: mark#(isNat(X)) -> active#(isNat(X))
p81: active#(U131(tt(),IL,M,N)) -> mark#(U132(isNatIListKind(IL),IL,M,N))
p82: mark#(U105(X1,X2)) -> active#(U105(mark(X1),X2))
p83: mark#(U104(X1,X2,X3)) -> active#(U104(mark(X1),X2,X3))
p84: active#(U12(tt(),V1)) -> mark#(U13(isNatList(V1)))
p85: mark#(isNatIListKind(X)) -> active#(isNatIListKind(X))
p86: active#(U114(tt(),L)) -> mark#(s(length(L)))
p87: mark#(U103(X1,X2,X3)) -> mark#(X1)
p88: mark#(U103(X1,X2,X3)) -> active#(U103(mark(X1),X2,X3))
p89: active#(U113(tt(),L,N)) -> mark#(U114(isNatKind(N),L))
p90: mark#(isNatKind(X)) -> active#(isNatKind(X))
p91: active#(U112(tt(),L,N)) -> mark#(U113(isNat(N),L,N))
p92: mark#(U102(X1,X2,X3)) -> mark#(X1)
p93: mark#(U101(X1,X2,X3)) -> active#(U101(mark(X1),X2,X3))
p94: active#(U111(tt(),L,N)) -> mark#(U112(isNatIListKind(L),L,N))
p95: active#(U11(tt(),V1)) -> mark#(U12(isNatIListKind(V1),V1))
p96: active#(U105(tt(),V2)) -> mark#(U106(isNatIList(V2)))
p97: active#(U104(tt(),V1,V2)) -> mark#(U105(isNat(V1),V2))
p98: active#(U103(tt(),V1,V2)) -> mark#(U104(isNatIListKind(V2),V1,V2))
p99: active#(U101(tt(),V1,V2)) -> mark#(U102(isNatKind(V1),V1,V2))

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19, p20, p21, p22, p23, p24, p25, p26, p27, p28, p29, p30, p31, p32, p33, p34, p35, p36, p37, p38, p39, p40, p41, p42, p43, p44, p45, p46, p47, p48, p49, p50, p51, p52, p53, p54, p55, p56, p57, p58, p59, p60, p61, p62, p63, p64, p65, p66, p67, p68, p69, p70, p71, p72, p73, p74, p75, p76, p77, p78, p79, p80, p81, p82, p83, p84, p85, p86, p87, p88, p89, p90, p91, p92, p93, p94, p95, p96, p97, p98, p99}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(U96(X)) -> mark#(X)
p2: mark#(U101(X1,X2,X3)) -> active#(U101(mark(X1),X2,X3))
p3: active#(U101(tt(),V1,V2)) -> mark#(U102(isNatKind(V1),V1,V2))
p4: mark#(U102(X1,X2,X3)) -> mark#(X1)
p5: mark#(isNatKind(X)) -> active#(isNatKind(X))
p6: active#(U103(tt(),V1,V2)) -> mark#(U104(isNatIListKind(V2),V1,V2))
p7: mark#(U103(X1,X2,X3)) -> active#(U103(mark(X1),X2,X3))
p8: active#(U104(tt(),V1,V2)) -> mark#(U105(isNat(V1),V2))
p9: mark#(U103(X1,X2,X3)) -> mark#(X1)
p10: mark#(isNatIListKind(X)) -> active#(isNatIListKind(X))
p11: active#(U105(tt(),V2)) -> mark#(U106(isNatIList(V2)))
p12: mark#(U104(X1,X2,X3)) -> active#(U104(mark(X1),X2,X3))
p13: active#(U11(tt(),V1)) -> mark#(U12(isNatIListKind(V1),V1))
p14: mark#(U105(X1,X2)) -> active#(U105(mark(X1),X2))
p15: active#(U111(tt(),L,N)) -> mark#(U112(isNatIListKind(L),L,N))
p16: mark#(isNat(X)) -> active#(isNat(X))
p17: active#(U112(tt(),L,N)) -> mark#(U113(isNat(N),L,N))
p18: mark#(U106(X)) -> mark#(X)
p19: mark#(isNatIList(X)) -> active#(isNatIList(X))
p20: active#(U113(tt(),L,N)) -> mark#(U114(isNatKind(N),L))
p21: mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2))
p22: active#(U114(tt(),L)) -> mark#(s(length(L)))
p23: mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2))
p24: active#(U12(tt(),V1)) -> mark#(U13(isNatList(V1)))
p25: mark#(U12(X1,X2)) -> mark#(X1)
p26: mark#(U111(X1,X2,X3)) -> active#(U111(mark(X1),X2,X3))
p27: active#(U131(tt(),IL,M,N)) -> mark#(U132(isNatIListKind(IL),IL,M,N))
p28: mark#(U112(X1,X2,X3)) -> active#(U112(mark(X1),X2,X3))
p29: active#(U132(tt(),IL,M,N)) -> mark#(U133(isNat(M),IL,M,N))
p30: mark#(U113(X1,X2,X3)) -> active#(U113(mark(X1),X2,X3))
p31: active#(U133(tt(),IL,M,N)) -> mark#(U134(isNatKind(M),IL,M,N))
p32: mark#(U114(X1,X2)) -> active#(U114(mark(X1),X2))
p33: active#(U134(tt(),IL,M,N)) -> mark#(U135(isNat(N),IL,M,N))
p34: mark#(s(X)) -> mark#(X)
p35: mark#(length(X)) -> active#(length(mark(X)))
p36: active#(U135(tt(),IL,M,N)) -> mark#(U136(isNatKind(N),IL,M,N))
p37: mark#(U13(X)) -> mark#(X)
p38: mark#(isNatList(X)) -> active#(isNatList(X))
p39: active#(U136(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL)))
p40: mark#(U131(X1,X2,X3,X4)) -> active#(U131(mark(X1),X2,X3,X4))
p41: active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1))
p42: mark#(U132(X1,X2,X3,X4)) -> active#(U132(mark(X1),X2,X3,X4))
p43: active#(U22(tt(),V1)) -> mark#(U23(isNat(V1)))
p44: mark#(U133(X1,X2,X3,X4)) -> active#(U133(mark(X1),X2,X3,X4))
p45: active#(U32(tt(),V)) -> mark#(U33(isNatList(V)))
p46: mark#(U134(X1,X2,X3,X4)) -> active#(U134(mark(X1),X2,X3,X4))
p47: active#(U41(tt(),V1,V2)) -> mark#(U42(isNatKind(V1),V1,V2))
p48: mark#(U135(X1,X2,X3,X4)) -> active#(U135(mark(X1),X2,X3,X4))
p49: active#(U42(tt(),V1,V2)) -> mark#(U43(isNatIListKind(V2),V1,V2))
p50: mark#(U136(X1,X2,X3,X4)) -> active#(U136(mark(X1),X2,X3,X4))
p51: active#(U43(tt(),V1,V2)) -> mark#(U44(isNatIListKind(V2),V1,V2))
p52: mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2))
p53: active#(U44(tt(),V1,V2)) -> mark#(U45(isNat(V1),V2))
p54: mark#(U21(X1,X2)) -> mark#(X1)
p55: mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2))
p56: active#(U45(tt(),V2)) -> mark#(U46(isNatIList(V2)))
p57: mark#(U22(X1,X2)) -> mark#(X1)
p58: mark#(U23(X)) -> mark#(X)
p59: mark#(U32(X1,X2)) -> active#(U32(mark(X1),X2))
p60: active#(U51(tt(),V2)) -> mark#(U52(isNatIListKind(V2)))
p61: mark#(U32(X1,X2)) -> mark#(X1)
p62: mark#(U33(X)) -> mark#(X)
p63: mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3))
p64: active#(U61(tt(),V2)) -> mark#(U62(isNatIListKind(V2)))
p65: mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3))
p66: active#(U91(tt(),V1,V2)) -> mark#(U92(isNatKind(V1),V1,V2))
p67: mark#(U42(X1,X2,X3)) -> mark#(X1)
p68: mark#(U43(X1,X2,X3)) -> active#(U43(mark(X1),X2,X3))
p69: active#(U92(tt(),V1,V2)) -> mark#(U93(isNatIListKind(V2),V1,V2))
p70: mark#(U44(X1,X2,X3)) -> active#(U44(mark(X1),X2,X3))
p71: active#(U93(tt(),V1,V2)) -> mark#(U94(isNatIListKind(V2),V1,V2))
p72: mark#(U45(X1,X2)) -> active#(U45(mark(X1),X2))
p73: active#(U94(tt(),V1,V2)) -> mark#(U95(isNat(V1),V2))
p74: mark#(U45(X1,X2)) -> mark#(X1)
p75: mark#(U46(X)) -> mark#(X)
p76: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p77: active#(U95(tt(),V2)) -> mark#(U96(isNatList(V2)))
p78: mark#(U51(X1,X2)) -> mark#(X1)
p79: mark#(U52(X)) -> mark#(X)
p80: mark#(U61(X1,X2)) -> active#(U61(mark(X1),X2))
p81: active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1))
p82: mark#(U62(X)) -> mark#(X)
p83: mark#(U71(X)) -> mark#(X)
p84: mark#(U81(X)) -> mark#(X)
p85: mark#(U91(X1,X2,X3)) -> active#(U91(mark(X1),X2,X3))
p86: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNatKind(V1),V1,V2))
p87: mark#(U91(X1,X2,X3)) -> mark#(X1)
p88: mark#(U92(X1,X2,X3)) -> active#(U92(mark(X1),X2,X3))
p89: active#(isNatIListKind(cons(V1,V2))) -> mark#(U51(isNatKind(V1),V2))
p90: mark#(U92(X1,X2,X3)) -> mark#(X1)
p91: mark#(U93(X1,X2,X3)) -> active#(U93(mark(X1),X2,X3))
p92: active#(isNatKind(length(V1))) -> mark#(U71(isNatIListKind(V1)))
p93: mark#(U94(X1,X2,X3)) -> active#(U94(mark(X1),X2,X3))
p94: active#(isNatKind(s(V1))) -> mark#(U81(isNatKind(V1)))
p95: mark#(U95(X1,X2)) -> active#(U95(mark(X1),X2))
p96: active#(isNatList(cons(V1,V2))) -> mark#(U91(isNatKind(V1),V1,V2))
p97: mark#(U95(X1,X2)) -> mark#(X1)
p98: active#(isNatList(take(V1,V2))) -> mark#(U101(isNatKind(V1),V1,V2))
p99: active#(length(cons(N,L))) -> mark#(U111(isNatList(L),L,N))

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41, r42, r43, r44, r45, r46, r47, r48, r49, r50, r51, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76, r77, r78, r79, r80, r81, r82, r83, r84, r85, r86, r87, r88, r89, r90, r91, r92, r93, r94, r95, r96, r97, r98, r99, r100, r101, r102, r103, r104, r105, r106, r107, r108, r109, r110, r111, r112, r113, r114, r115, r116, r117, r118, r119, r120, r121, r122, r123, r124, r125, r126, r127, r128, r129, r130, r131, r132, r133, r134, r135, r136, r137, r138, r139, r140, r141, r142, r143, r144, r145, r146, r147, r148, r149, r150, r151, r152, r153, r154, r155, r156, r157, r158, r159, r160, r161, r162, r163, r164, r165, r166, r167, r168, r169, r170, r171, r172, r173, r174, r175, r176, r177, r178, r179, r180, r181, r182, r183, r184, r185, r186, r187, r188, r189, r190, r191, r192, r193, r194, r195, r196, r197, r198, r199, r200, r201, r202, r203, r204, r205, r206, r207, r208, r209, r210, r211, r212, r213, r214, r215, r216, r217, r218, r219, r220, r221, r222, r223, r224, r225, r226, r227, r228, r229, r230, r231, r232, r233, r234, r235, r236, r237, r238, r239, r240, r241, r242, r243, r244, r245, r246, r247, r248, r249, r250, r251, r252, r253, r254, r255, r256, r257, r258, r259, r260, r261, r262, r263, r264, r265, r266, r267, r268, r269, r270, r271, r272, r273, r274, r275, r276, r277, r278, r279, r280, r281, r282, r283, r284, r285, r286, r287, r288, r289, r290, r291, r292, r293, r294, r295, r296, r297, r298, r299, r300, r301, r302, r303, r304, r305, r306, r307, r308, r309, r310, r311, r312, r313, r314, r315, r316, r317, r318, r319, r320, r321, r322, r323, r324, r325, r326, r327, r328, r329, r330, r331, r332, r333, r334, r335, r336, r337, r338, r339, r340, r341, r342, r343, r344, r345, r346, r347, r348, r349, r350, r351, r352, r353, r354, r355, r356, r357, r358

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    mark#_A(x1) = x1 + 1
    U96_A(x1) = max{68, x1 + 22}
    U101_A(x1,x2,x3) = max{270, x1 - 2, x2 + 25, x3 + 141}
    active#_A(x1) = max{23, x1 + 1}
    mark_A(x1) = max{26, x1}
    tt_A = 68
    U102_A(x1,x2,x3) = max{x1 + 1, x2 + 25, x3 + 140}
    isNatKind_A(x1) = max{64, x1 + 21}
    U103_A(x1,x2,x3) = max{x1 + 26, x2 + 25, x3 + 140}
    U104_A(x1,x2,x3) = max{x1 - 65, x2 - 2, x3 + 139}
    isNatIListKind_A(x1) = max{86, x1 + 67}
    U105_A(x1,x2) = max{139, x2 + 138}
    isNat_A(x1) = max{33, x1 + 21}
    U106_A(x1) = max{138, x1 + 26}
    isNatIList_A(x1) = x1 + 112
    U11_A(x1,x2) = max{113, x1 + 87, x2 + 93}
    U12_A(x1,x2) = max{x1 + 26, x2 + 93}
    U111_A(x1,x2,x3) = max{x1 + 136, x2 + 178}
    U112_A(x1,x2,x3) = max{204, x1 - 70, x2 + 178}
    U113_A(x1,x2,x3) = max{204, x2 + 178}
    U114_A(x1,x2) = max{204, x2 + 178}
    s_A(x1) = max{204, x1 + 26}
    length_A(x1) = max{178, x1 + 152}
    U13_A(x1) = max{94, x1 + 51}
    isNatList_A(x1) = x1 + 42
    U131_A(x1,x2,x3,x4) = max{269, x1 + 45, x2 + 202, x3 + 228, x4 + 155}
    U132_A(x1,x2,x3,x4) = max{268, x1 + 26, x2 + 202, x3 + 228, x4 + 154}
    U133_A(x1,x2,x3,x4) = max{268, x1 + 60, x2 + 202, x3 + 228, x4 + 154}
    U134_A(x1,x2,x3,x4) = max{267, x1 + 63, x2 + 202, x3 + 228, x4 + 154}
    U135_A(x1,x2,x3,x4) = max{266, x2 + 202, x3 + 228, x4 + 153}
    U136_A(x1,x2,x3,x4) = max{265, x2 + 202, x3 + 228, x4 + 153}
    cons_A(x1,x2) = max{x1 - 21, x2 + 26}
    take_A(x1,x2) = max{228, x1 + 202, x2 + 176}
    U21_A(x1,x2) = max{x1 + 21, x2 + 47}
    U22_A(x1,x2) = max{x1 + 21, x2 + 47}
    U23_A(x1) = max{89, x1 + 26}
    U32_A(x1,x2) = max{x1 + 26, x2 + 69}
    U33_A(x1) = max{68, x1 + 27}
    U41_A(x1,x2,x3) = max{x1 + 26, x2 + 91, x3 + 138}
    U42_A(x1,x2,x3) = max{x1 + 26, x2 + 82, x3 + 138}
    U43_A(x1,x2,x3) = max{x1 + 52, x2 + 81, x3 + 138}
    U44_A(x1,x2,x3) = max{x1 - 6, x2 + 42, x3 + 138}
    U45_A(x1,x2) = max{x1 + 21, x2 + 138}
    U46_A(x1) = max{88, x1 + 26}
    U51_A(x1,x2) = max{x1 + 23, x2 + 92}
    U52_A(x1) = max{90, x1 + 6}
    U61_A(x1,x2) = max{x1 + 44, x2 + 112}
    U62_A(x1) = max{85, x1 + 26}
    U91_A(x1,x2,x3) = max{x1, x2 + 21, x3 + 68}
    U92_A(x1,x2,x3) = max{x1, x2 + 21, x3 + 68}
    U93_A(x1,x2,x3) = max{x2 + 21, x3 + 68}
    U94_A(x1,x2,x3) = max{x2 + 21, x3 + 64}
    U95_A(x1,x2) = max{x1, x2 + 64}
    U71_A(x1) = max{68, x1 + 26}
    U81_A(x1) = max{68, x1 + 26}
    active_A(x1) = max{26, x1}
    zeros_A = 0
    |0|_A = 47
    U121_A(x1,x2) = max{x1 + 44, x2 + 112}
    U122_A(x1) = max{52, x1 + 26}
    nil_A = 27
    U31_A(x1,x2) = max{x1 + 25, x2 + 112}

The next rules are strictly ordered:

  p1, p4, p6, p9, p18, p25, p34, p37, p54, p57, p58, p61, p62, p67, p74, p75, p78, p79, p82, p83, p84, p89, p92

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(U101(X1,X2,X3)) -> active#(U101(mark(X1),X2,X3))
p2: active#(U101(tt(),V1,V2)) -> mark#(U102(isNatKind(V1),V1,V2))
p3: mark#(isNatKind(X)) -> active#(isNatKind(X))
p4: mark#(U103(X1,X2,X3)) -> active#(U103(mark(X1),X2,X3))
p5: active#(U104(tt(),V1,V2)) -> mark#(U105(isNat(V1),V2))
p6: mark#(isNatIListKind(X)) -> active#(isNatIListKind(X))
p7: active#(U105(tt(),V2)) -> mark#(U106(isNatIList(V2)))
p8: mark#(U104(X1,X2,X3)) -> active#(U104(mark(X1),X2,X3))
p9: active#(U11(tt(),V1)) -> mark#(U12(isNatIListKind(V1),V1))
p10: mark#(U105(X1,X2)) -> active#(U105(mark(X1),X2))
p11: active#(U111(tt(),L,N)) -> mark#(U112(isNatIListKind(L),L,N))
p12: mark#(isNat(X)) -> active#(isNat(X))
p13: active#(U112(tt(),L,N)) -> mark#(U113(isNat(N),L,N))
p14: mark#(isNatIList(X)) -> active#(isNatIList(X))
p15: active#(U113(tt(),L,N)) -> mark#(U114(isNatKind(N),L))
p16: mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2))
p17: active#(U114(tt(),L)) -> mark#(s(length(L)))
p18: mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2))
p19: active#(U12(tt(),V1)) -> mark#(U13(isNatList(V1)))
p20: mark#(U111(X1,X2,X3)) -> active#(U111(mark(X1),X2,X3))
p21: active#(U131(tt(),IL,M,N)) -> mark#(U132(isNatIListKind(IL),IL,M,N))
p22: mark#(U112(X1,X2,X3)) -> active#(U112(mark(X1),X2,X3))
p23: active#(U132(tt(),IL,M,N)) -> mark#(U133(isNat(M),IL,M,N))
p24: mark#(U113(X1,X2,X3)) -> active#(U113(mark(X1),X2,X3))
p25: active#(U133(tt(),IL,M,N)) -> mark#(U134(isNatKind(M),IL,M,N))
p26: mark#(U114(X1,X2)) -> active#(U114(mark(X1),X2))
p27: active#(U134(tt(),IL,M,N)) -> mark#(U135(isNat(N),IL,M,N))
p28: mark#(length(X)) -> active#(length(mark(X)))
p29: active#(U135(tt(),IL,M,N)) -> mark#(U136(isNatKind(N),IL,M,N))
p30: mark#(isNatList(X)) -> active#(isNatList(X))
p31: active#(U136(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL)))
p32: mark#(U131(X1,X2,X3,X4)) -> active#(U131(mark(X1),X2,X3,X4))
p33: active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1))
p34: mark#(U132(X1,X2,X3,X4)) -> active#(U132(mark(X1),X2,X3,X4))
p35: active#(U22(tt(),V1)) -> mark#(U23(isNat(V1)))
p36: mark#(U133(X1,X2,X3,X4)) -> active#(U133(mark(X1),X2,X3,X4))
p37: active#(U32(tt(),V)) -> mark#(U33(isNatList(V)))
p38: mark#(U134(X1,X2,X3,X4)) -> active#(U134(mark(X1),X2,X3,X4))
p39: active#(U41(tt(),V1,V2)) -> mark#(U42(isNatKind(V1),V1,V2))
p40: mark#(U135(X1,X2,X3,X4)) -> active#(U135(mark(X1),X2,X3,X4))
p41: active#(U42(tt(),V1,V2)) -> mark#(U43(isNatIListKind(V2),V1,V2))
p42: mark#(U136(X1,X2,X3,X4)) -> active#(U136(mark(X1),X2,X3,X4))
p43: active#(U43(tt(),V1,V2)) -> mark#(U44(isNatIListKind(V2),V1,V2))
p44: mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2))
p45: active#(U44(tt(),V1,V2)) -> mark#(U45(isNat(V1),V2))
p46: mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2))
p47: active#(U45(tt(),V2)) -> mark#(U46(isNatIList(V2)))
p48: mark#(U32(X1,X2)) -> active#(U32(mark(X1),X2))
p49: active#(U51(tt(),V2)) -> mark#(U52(isNatIListKind(V2)))
p50: mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3))
p51: active#(U61(tt(),V2)) -> mark#(U62(isNatIListKind(V2)))
p52: mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3))
p53: active#(U91(tt(),V1,V2)) -> mark#(U92(isNatKind(V1),V1,V2))
p54: mark#(U43(X1,X2,X3)) -> active#(U43(mark(X1),X2,X3))
p55: active#(U92(tt(),V1,V2)) -> mark#(U93(isNatIListKind(V2),V1,V2))
p56: mark#(U44(X1,X2,X3)) -> active#(U44(mark(X1),X2,X3))
p57: active#(U93(tt(),V1,V2)) -> mark#(U94(isNatIListKind(V2),V1,V2))
p58: mark#(U45(X1,X2)) -> active#(U45(mark(X1),X2))
p59: active#(U94(tt(),V1,V2)) -> mark#(U95(isNat(V1),V2))
p60: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p61: active#(U95(tt(),V2)) -> mark#(U96(isNatList(V2)))
p62: mark#(U61(X1,X2)) -> active#(U61(mark(X1),X2))
p63: active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1))
p64: mark#(U91(X1,X2,X3)) -> active#(U91(mark(X1),X2,X3))
p65: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNatKind(V1),V1,V2))
p66: mark#(U91(X1,X2,X3)) -> mark#(X1)
p67: mark#(U92(X1,X2,X3)) -> active#(U92(mark(X1),X2,X3))
p68: mark#(U92(X1,X2,X3)) -> mark#(X1)
p69: mark#(U93(X1,X2,X3)) -> active#(U93(mark(X1),X2,X3))
p70: mark#(U94(X1,X2,X3)) -> active#(U94(mark(X1),X2,X3))
p71: active#(isNatKind(s(V1))) -> mark#(U81(isNatKind(V1)))
p72: mark#(U95(X1,X2)) -> active#(U95(mark(X1),X2))
p73: active#(isNatList(cons(V1,V2))) -> mark#(U91(isNatKind(V1),V1,V2))
p74: mark#(U95(X1,X2)) -> mark#(X1)
p75: active#(isNatList(take(V1,V2))) -> mark#(U101(isNatKind(V1),V1,V2))
p76: active#(length(cons(N,L))) -> mark#(U111(isNatList(L),L,N))

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19, p20, p21, p22, p23, p24, p25, p26, p27, p28, p29, p30, p31, p32, p33, p34, p35, p36, p37, p38, p39, p40, p41, p42, p43, p44, p45, p46, p47, p48, p49, p50, p51, p52, p53, p54, p55, p56, p57, p58, p59, p60, p61, p62, p63, p64, p65, p66, p67, p68, p69, p70, p71, p72, p73, p74, p75, p76}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(U101(X1,X2,X3)) -> active#(U101(mark(X1),X2,X3))
p2: active#(length(cons(N,L))) -> mark#(U111(isNatList(L),L,N))
p3: mark#(U95(X1,X2)) -> mark#(X1)
p4: mark#(U95(X1,X2)) -> active#(U95(mark(X1),X2))
p5: active#(isNatList(take(V1,V2))) -> mark#(U101(isNatKind(V1),V1,V2))
p6: mark#(U94(X1,X2,X3)) -> active#(U94(mark(X1),X2,X3))
p7: active#(isNatList(cons(V1,V2))) -> mark#(U91(isNatKind(V1),V1,V2))
p8: mark#(U93(X1,X2,X3)) -> active#(U93(mark(X1),X2,X3))
p9: active#(isNatKind(s(V1))) -> mark#(U81(isNatKind(V1)))
p10: mark#(U92(X1,X2,X3)) -> mark#(X1)
p11: mark#(U92(X1,X2,X3)) -> active#(U92(mark(X1),X2,X3))
p12: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNatKind(V1),V1,V2))
p13: mark#(U91(X1,X2,X3)) -> mark#(X1)
p14: mark#(U91(X1,X2,X3)) -> active#(U91(mark(X1),X2,X3))
p15: active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1))
p16: mark#(U61(X1,X2)) -> active#(U61(mark(X1),X2))
p17: active#(U95(tt(),V2)) -> mark#(U96(isNatList(V2)))
p18: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p19: active#(U94(tt(),V1,V2)) -> mark#(U95(isNat(V1),V2))
p20: mark#(U45(X1,X2)) -> active#(U45(mark(X1),X2))
p21: active#(U93(tt(),V1,V2)) -> mark#(U94(isNatIListKind(V2),V1,V2))
p22: mark#(U44(X1,X2,X3)) -> active#(U44(mark(X1),X2,X3))
p23: active#(U92(tt(),V1,V2)) -> mark#(U93(isNatIListKind(V2),V1,V2))
p24: mark#(U43(X1,X2,X3)) -> active#(U43(mark(X1),X2,X3))
p25: active#(U91(tt(),V1,V2)) -> mark#(U92(isNatKind(V1),V1,V2))
p26: mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3))
p27: active#(U61(tt(),V2)) -> mark#(U62(isNatIListKind(V2)))
p28: mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3))
p29: active#(U51(tt(),V2)) -> mark#(U52(isNatIListKind(V2)))
p30: mark#(U32(X1,X2)) -> active#(U32(mark(X1),X2))
p31: active#(U45(tt(),V2)) -> mark#(U46(isNatIList(V2)))
p32: mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2))
p33: active#(U44(tt(),V1,V2)) -> mark#(U45(isNat(V1),V2))
p34: mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2))
p35: active#(U43(tt(),V1,V2)) -> mark#(U44(isNatIListKind(V2),V1,V2))
p36: mark#(U136(X1,X2,X3,X4)) -> active#(U136(mark(X1),X2,X3,X4))
p37: active#(U42(tt(),V1,V2)) -> mark#(U43(isNatIListKind(V2),V1,V2))
p38: mark#(U135(X1,X2,X3,X4)) -> active#(U135(mark(X1),X2,X3,X4))
p39: active#(U41(tt(),V1,V2)) -> mark#(U42(isNatKind(V1),V1,V2))
p40: mark#(U134(X1,X2,X3,X4)) -> active#(U134(mark(X1),X2,X3,X4))
p41: active#(U32(tt(),V)) -> mark#(U33(isNatList(V)))
p42: mark#(U133(X1,X2,X3,X4)) -> active#(U133(mark(X1),X2,X3,X4))
p43: active#(U22(tt(),V1)) -> mark#(U23(isNat(V1)))
p44: mark#(U132(X1,X2,X3,X4)) -> active#(U132(mark(X1),X2,X3,X4))
p45: active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1))
p46: mark#(U131(X1,X2,X3,X4)) -> active#(U131(mark(X1),X2,X3,X4))
p47: active#(U136(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL)))
p48: mark#(isNatList(X)) -> active#(isNatList(X))
p49: active#(U135(tt(),IL,M,N)) -> mark#(U136(isNatKind(N),IL,M,N))
p50: mark#(length(X)) -> active#(length(mark(X)))
p51: active#(U134(tt(),IL,M,N)) -> mark#(U135(isNat(N),IL,M,N))
p52: mark#(U114(X1,X2)) -> active#(U114(mark(X1),X2))
p53: active#(U133(tt(),IL,M,N)) -> mark#(U134(isNatKind(M),IL,M,N))
p54: mark#(U113(X1,X2,X3)) -> active#(U113(mark(X1),X2,X3))
p55: active#(U132(tt(),IL,M,N)) -> mark#(U133(isNat(M),IL,M,N))
p56: mark#(U112(X1,X2,X3)) -> active#(U112(mark(X1),X2,X3))
p57: active#(U131(tt(),IL,M,N)) -> mark#(U132(isNatIListKind(IL),IL,M,N))
p58: mark#(U111(X1,X2,X3)) -> active#(U111(mark(X1),X2,X3))
p59: active#(U12(tt(),V1)) -> mark#(U13(isNatList(V1)))
p60: mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2))
p61: active#(U114(tt(),L)) -> mark#(s(length(L)))
p62: mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2))
p63: active#(U113(tt(),L,N)) -> mark#(U114(isNatKind(N),L))
p64: mark#(isNatIList(X)) -> active#(isNatIList(X))
p65: active#(U112(tt(),L,N)) -> mark#(U113(isNat(N),L,N))
p66: mark#(isNat(X)) -> active#(isNat(X))
p67: active#(U111(tt(),L,N)) -> mark#(U112(isNatIListKind(L),L,N))
p68: mark#(U105(X1,X2)) -> active#(U105(mark(X1),X2))
p69: active#(U11(tt(),V1)) -> mark#(U12(isNatIListKind(V1),V1))
p70: mark#(U104(X1,X2,X3)) -> active#(U104(mark(X1),X2,X3))
p71: active#(U105(tt(),V2)) -> mark#(U106(isNatIList(V2)))
p72: mark#(isNatIListKind(X)) -> active#(isNatIListKind(X))
p73: active#(U104(tt(),V1,V2)) -> mark#(U105(isNat(V1),V2))
p74: mark#(U103(X1,X2,X3)) -> active#(U103(mark(X1),X2,X3))
p75: active#(U101(tt(),V1,V2)) -> mark#(U102(isNatKind(V1),V1,V2))
p76: mark#(isNatKind(X)) -> active#(isNatKind(X))

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41, r42, r43, r44, r45, r46, r47, r48, r49, r50, r51, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76, r77, r78, r79, r80, r81, r82, r83, r84, r85, r86, r87, r88, r89, r90, r91, r92, r93, r94, r95, r96, r97, r98, r99, r100, r101, r102, r103, r104, r105, r106, r107, r108, r109, r110, r111, r112, r113, r114, r115, r116, r117, r118, r119, r120, r121, r122, r123, r124, r125, r126, r127, r128, r129, r130, r131, r132, r133, r134, r135, r136, r137, r138, r139, r140, r141, r142, r143, r144, r145, r146, r147, r148, r149, r150, r151, r152, r153, r154, r155, r156, r157, r158, r159, r160, r161, r162, r163, r164, r165, r166, r167, r168, r169, r170, r171, r172, r173, r174, r175, r176, r177, r178, r179, r180, r181, r182, r183, r184, r185, r186, r187, r188, r189, r190, r191, r192, r193, r194, r195, r196, r197, r198, r199, r200, r201, r202, r203, r204, r205, r206, r207, r208, r209, r210, r211, r212, r213, r214, r215, r216, r217, r218, r219, r220, r221, r222, r223, r224, r225, r226, r227, r228, r229, r230, r231, r232, r233, r234, r235, r236, r237, r238, r239, r240, r241, r242, r243, r244, r245, r246, r247, r248, r249, r250, r251, r252, r253, r254, r255, r256, r257, r258, r259, r260, r261, r262, r263, r264, r265, r266, r267, r268, r269, r270, r271, r272, r273, r274, r275, r276, r277, r278, r279, r280, r281, r282, r283, r284, r285, r286, r287, r288, r289, r290, r291, r292, r293, r294, r295, r296, r297, r298, r299, r300, r301, r302, r303, r304, r305, r306, r307, r308, r309, r310, r311, r312, r313, r314, r315, r316, r317, r318, r319, r320, r321, r322, r323, r324, r325, r326, r327, r328, r329, r330, r331, r332, r333, r334, r335, r336, r337, r338, r339, r340, r341, r342, r343, r344, r345, r346, r347, r348, r349, r350, r351, r352, r353, r354, r355, r356, r357, r358

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    mark#_A(x1) = max{6, x1 + 1}
    U101_A(x1,x2,x3) = max{x1 + 528, x2 + 559, x3 + 559}
    active#_A(x1) = x1 + 1
    mark_A(x1) = max{31, x1}
    length_A(x1) = max{203, x1 + 171}
    cons_A(x1,x2) = max{111, x1 + 80, x2 - 12}
    U111_A(x1,x2,x3) = max{x1 + 141, x2 + 159, x3 + 251}
    isNatList_A(x1) = x1 + 18
    U95_A(x1,x2) = max{32, x1}
    take_A(x1,x2) = max{583, x1 + 552, x2 + 552}
    isNatKind_A(x1) = x1 + 42
    U94_A(x1,x2,x3) = x2 + 98
    U91_A(x1,x2,x3) = max{x1, x2 + 98}
    U93_A(x1,x2,x3) = x2 + 98
    s_A(x1) = max{18, x1 - 12}
    U81_A(x1) = 60
    U92_A(x1,x2,x3) = max{x1, x2 + 98}
    isNatIList_A(x1) = x1 + 321
    U41_A(x1,x2,x3) = max{x1 + 56, x2 + 401, x3 + 309}
    isNat_A(x1) = x1 + 98
    U21_A(x1,x2) = max{x1 + 44, x2 + 75}
    U61_A(x1,x2) = max{x1 + 10, x2 + 146}
    tt_A = 9
    U96_A(x1) = 32
    U51_A(x1,x2) = max{x1 + 17, x2 + 48}
    U45_A(x1,x2) = max{x1 + 89, x2 + 309}
    isNatIListKind_A(x1) = x1 + 129
    U44_A(x1,x2,x3) = max{x1 + 179, x2 + 211, x3 + 309}
    U43_A(x1,x2,x3) = max{x1 + 78, x2 + 211, x3 + 309}
    U42_A(x1,x2,x3) = max{x1 + 67, x2 + 211, x3 + 309}
    U62_A(x1) = max{146, x1 + 16}
    U52_A(x1) = 48
    U32_A(x1,x2) = max{x1 + 33, x2 + 64}
    U46_A(x1) = max{309, x1 - 13}
    U22_A(x1,x2) = max{x1 + 33, x2 + 64}
    U136_A(x1,x2,x3,x4) = max{x1 + 563, x2 + 540, x3 + 540, x4 + 594}
    U135_A(x1,x2,x3,x4) = max{x1 + 16, x2 + 540, x3 + 540, x4 + 606}
    U134_A(x1,x2,x3,x4) = max{x1 + 497, x2 + 540, x3 + 540, x4 + 606}
    U33_A(x1) = 64
    U133_A(x1,x2,x3,x4) = max{x1 + 442, x2 + 540, x3 + 540, x4 + 606}
    U23_A(x1) = max{64, x1 - 64}
    U132_A(x1,x2,x3,x4) = max{x1 + 20, x2 + 540, x3 + 540, x4 + 606}
    U131_A(x1,x2,x3,x4) = max{x1 + 22, x2 + 540, x3 + 540, x4 + 607}
    U114_A(x1,x2) = max{240, x1 + 209, x2 + 159}
    U113_A(x1,x2,x3) = max{x1 + 60, x2 + 159, x3 + 251}
    U112_A(x1,x2,x3) = max{x1 + 29, x2 + 159, x3 + 251}
    U12_A(x1,x2) = max{x1 + 51, x2 + 82}
    U13_A(x1) = 32
    U11_A(x1,x2) = max{x1 + 22, x2 + 180}
    U105_A(x1,x2) = max{438, x1 + 335, x2 + 343}
    U104_A(x1,x2,x3) = max{x1 + 430, x2 + 433, x3 + 461}
    U106_A(x1) = max{53, x1 + 22}
    U103_A(x1,x2,x3) = max{x1 + 401, x2 + 433, x3 + 559}
    U102_A(x1,x2,x3) = max{x1 + 507, x2 + 507, x3 + 559}
    active_A(x1) = max{17, x1}
    zeros_A = 265
    |0|_A = 185
    U121_A(x1,x2) = max{584, x2 + 552}
    U122_A(x1) = max{553, x1 + 423}
    nil_A = 13
    U31_A(x1,x2) = max{x1 + 14, x2 + 162}
    U71_A(x1) = max{214, x1 + 16}

The next rules are strictly ordered:

  p59

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(U101(X1,X2,X3)) -> active#(U101(mark(X1),X2,X3))
p2: active#(length(cons(N,L))) -> mark#(U111(isNatList(L),L,N))
p3: mark#(U95(X1,X2)) -> mark#(X1)
p4: mark#(U95(X1,X2)) -> active#(U95(mark(X1),X2))
p5: active#(isNatList(take(V1,V2))) -> mark#(U101(isNatKind(V1),V1,V2))
p6: mark#(U94(X1,X2,X3)) -> active#(U94(mark(X1),X2,X3))
p7: active#(isNatList(cons(V1,V2))) -> mark#(U91(isNatKind(V1),V1,V2))
p8: mark#(U93(X1,X2,X3)) -> active#(U93(mark(X1),X2,X3))
p9: active#(isNatKind(s(V1))) -> mark#(U81(isNatKind(V1)))
p10: mark#(U92(X1,X2,X3)) -> mark#(X1)
p11: mark#(U92(X1,X2,X3)) -> active#(U92(mark(X1),X2,X3))
p12: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNatKind(V1),V1,V2))
p13: mark#(U91(X1,X2,X3)) -> mark#(X1)
p14: mark#(U91(X1,X2,X3)) -> active#(U91(mark(X1),X2,X3))
p15: active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1))
p16: mark#(U61(X1,X2)) -> active#(U61(mark(X1),X2))
p17: active#(U95(tt(),V2)) -> mark#(U96(isNatList(V2)))
p18: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p19: active#(U94(tt(),V1,V2)) -> mark#(U95(isNat(V1),V2))
p20: mark#(U45(X1,X2)) -> active#(U45(mark(X1),X2))
p21: active#(U93(tt(),V1,V2)) -> mark#(U94(isNatIListKind(V2),V1,V2))
p22: mark#(U44(X1,X2,X3)) -> active#(U44(mark(X1),X2,X3))
p23: active#(U92(tt(),V1,V2)) -> mark#(U93(isNatIListKind(V2),V1,V2))
p24: mark#(U43(X1,X2,X3)) -> active#(U43(mark(X1),X2,X3))
p25: active#(U91(tt(),V1,V2)) -> mark#(U92(isNatKind(V1),V1,V2))
p26: mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3))
p27: active#(U61(tt(),V2)) -> mark#(U62(isNatIListKind(V2)))
p28: mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3))
p29: active#(U51(tt(),V2)) -> mark#(U52(isNatIListKind(V2)))
p30: mark#(U32(X1,X2)) -> active#(U32(mark(X1),X2))
p31: active#(U45(tt(),V2)) -> mark#(U46(isNatIList(V2)))
p32: mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2))
p33: active#(U44(tt(),V1,V2)) -> mark#(U45(isNat(V1),V2))
p34: mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2))
p35: active#(U43(tt(),V1,V2)) -> mark#(U44(isNatIListKind(V2),V1,V2))
p36: mark#(U136(X1,X2,X3,X4)) -> active#(U136(mark(X1),X2,X3,X4))
p37: active#(U42(tt(),V1,V2)) -> mark#(U43(isNatIListKind(V2),V1,V2))
p38: mark#(U135(X1,X2,X3,X4)) -> active#(U135(mark(X1),X2,X3,X4))
p39: active#(U41(tt(),V1,V2)) -> mark#(U42(isNatKind(V1),V1,V2))
p40: mark#(U134(X1,X2,X3,X4)) -> active#(U134(mark(X1),X2,X3,X4))
p41: active#(U32(tt(),V)) -> mark#(U33(isNatList(V)))
p42: mark#(U133(X1,X2,X3,X4)) -> active#(U133(mark(X1),X2,X3,X4))
p43: active#(U22(tt(),V1)) -> mark#(U23(isNat(V1)))
p44: mark#(U132(X1,X2,X3,X4)) -> active#(U132(mark(X1),X2,X3,X4))
p45: active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1))
p46: mark#(U131(X1,X2,X3,X4)) -> active#(U131(mark(X1),X2,X3,X4))
p47: active#(U136(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL)))
p48: mark#(isNatList(X)) -> active#(isNatList(X))
p49: active#(U135(tt(),IL,M,N)) -> mark#(U136(isNatKind(N),IL,M,N))
p50: mark#(length(X)) -> active#(length(mark(X)))
p51: active#(U134(tt(),IL,M,N)) -> mark#(U135(isNat(N),IL,M,N))
p52: mark#(U114(X1,X2)) -> active#(U114(mark(X1),X2))
p53: active#(U133(tt(),IL,M,N)) -> mark#(U134(isNatKind(M),IL,M,N))
p54: mark#(U113(X1,X2,X3)) -> active#(U113(mark(X1),X2,X3))
p55: active#(U132(tt(),IL,M,N)) -> mark#(U133(isNat(M),IL,M,N))
p56: mark#(U112(X1,X2,X3)) -> active#(U112(mark(X1),X2,X3))
p57: active#(U131(tt(),IL,M,N)) -> mark#(U132(isNatIListKind(IL),IL,M,N))
p58: mark#(U111(X1,X2,X3)) -> active#(U111(mark(X1),X2,X3))
p59: mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2))
p60: active#(U114(tt(),L)) -> mark#(s(length(L)))
p61: mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2))
p62: active#(U113(tt(),L,N)) -> mark#(U114(isNatKind(N),L))
p63: mark#(isNatIList(X)) -> active#(isNatIList(X))
p64: active#(U112(tt(),L,N)) -> mark#(U113(isNat(N),L,N))
p65: mark#(isNat(X)) -> active#(isNat(X))
p66: active#(U111(tt(),L,N)) -> mark#(U112(isNatIListKind(L),L,N))
p67: mark#(U105(X1,X2)) -> active#(U105(mark(X1),X2))
p68: active#(U11(tt(),V1)) -> mark#(U12(isNatIListKind(V1),V1))
p69: mark#(U104(X1,X2,X3)) -> active#(U104(mark(X1),X2,X3))
p70: active#(U105(tt(),V2)) -> mark#(U106(isNatIList(V2)))
p71: mark#(isNatIListKind(X)) -> active#(isNatIListKind(X))
p72: active#(U104(tt(),V1,V2)) -> mark#(U105(isNat(V1),V2))
p73: mark#(U103(X1,X2,X3)) -> active#(U103(mark(X1),X2,X3))
p74: active#(U101(tt(),V1,V2)) -> mark#(U102(isNatKind(V1),V1,V2))
p75: mark#(isNatKind(X)) -> active#(isNatKind(X))

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19, p20, p21, p22, p23, p24, p25, p26, p27, p28, p29, p30, p31, p32, p33, p34, p35, p36, p37, p38, p39, p40, p41, p42, p43, p44, p45, p46, p47, p48, p49, p50, p51, p52, p53, p54, p55, p56, p57, p58, p59, p60, p61, p62, p63, p64, p65, p66, p67, p68, p69, p70, p71, p72, p73, p74, p75}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(U101(X1,X2,X3)) -> active#(U101(mark(X1),X2,X3))
p2: active#(U101(tt(),V1,V2)) -> mark#(U102(isNatKind(V1),V1,V2))
p3: mark#(isNatKind(X)) -> active#(isNatKind(X))
p4: active#(U104(tt(),V1,V2)) -> mark#(U105(isNat(V1),V2))
p5: mark#(U103(X1,X2,X3)) -> active#(U103(mark(X1),X2,X3))
p6: active#(U105(tt(),V2)) -> mark#(U106(isNatIList(V2)))
p7: mark#(isNatIListKind(X)) -> active#(isNatIListKind(X))
p8: active#(U11(tt(),V1)) -> mark#(U12(isNatIListKind(V1),V1))
p9: mark#(U104(X1,X2,X3)) -> active#(U104(mark(X1),X2,X3))
p10: active#(U111(tt(),L,N)) -> mark#(U112(isNatIListKind(L),L,N))
p11: mark#(U105(X1,X2)) -> active#(U105(mark(X1),X2))
p12: active#(U112(tt(),L,N)) -> mark#(U113(isNat(N),L,N))
p13: mark#(isNat(X)) -> active#(isNat(X))
p14: active#(U113(tt(),L,N)) -> mark#(U114(isNatKind(N),L))
p15: mark#(isNatIList(X)) -> active#(isNatIList(X))
p16: active#(U114(tt(),L)) -> mark#(s(length(L)))
p17: mark#(U11(X1,X2)) -> active#(U11(mark(X1),X2))
p18: active#(U131(tt(),IL,M,N)) -> mark#(U132(isNatIListKind(IL),IL,M,N))
p19: mark#(U12(X1,X2)) -> active#(U12(mark(X1),X2))
p20: active#(U132(tt(),IL,M,N)) -> mark#(U133(isNat(M),IL,M,N))
p21: mark#(U111(X1,X2,X3)) -> active#(U111(mark(X1),X2,X3))
p22: active#(U133(tt(),IL,M,N)) -> mark#(U134(isNatKind(M),IL,M,N))
p23: mark#(U112(X1,X2,X3)) -> active#(U112(mark(X1),X2,X3))
p24: active#(U134(tt(),IL,M,N)) -> mark#(U135(isNat(N),IL,M,N))
p25: mark#(U113(X1,X2,X3)) -> active#(U113(mark(X1),X2,X3))
p26: active#(U135(tt(),IL,M,N)) -> mark#(U136(isNatKind(N),IL,M,N))
p27: mark#(U114(X1,X2)) -> active#(U114(mark(X1),X2))
p28: active#(U136(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL)))
p29: mark#(length(X)) -> active#(length(mark(X)))
p30: active#(U21(tt(),V1)) -> mark#(U22(isNatKind(V1),V1))
p31: mark#(isNatList(X)) -> active#(isNatList(X))
p32: active#(U22(tt(),V1)) -> mark#(U23(isNat(V1)))
p33: mark#(U131(X1,X2,X3,X4)) -> active#(U131(mark(X1),X2,X3,X4))
p34: active#(U32(tt(),V)) -> mark#(U33(isNatList(V)))
p35: mark#(U132(X1,X2,X3,X4)) -> active#(U132(mark(X1),X2,X3,X4))
p36: active#(U41(tt(),V1,V2)) -> mark#(U42(isNatKind(V1),V1,V2))
p37: mark#(U133(X1,X2,X3,X4)) -> active#(U133(mark(X1),X2,X3,X4))
p38: active#(U42(tt(),V1,V2)) -> mark#(U43(isNatIListKind(V2),V1,V2))
p39: mark#(U134(X1,X2,X3,X4)) -> active#(U134(mark(X1),X2,X3,X4))
p40: active#(U43(tt(),V1,V2)) -> mark#(U44(isNatIListKind(V2),V1,V2))
p41: mark#(U135(X1,X2,X3,X4)) -> active#(U135(mark(X1),X2,X3,X4))
p42: active#(U44(tt(),V1,V2)) -> mark#(U45(isNat(V1),V2))
p43: mark#(U136(X1,X2,X3,X4)) -> active#(U136(mark(X1),X2,X3,X4))
p44: active#(U45(tt(),V2)) -> mark#(U46(isNatIList(V2)))
p45: mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2))
p46: active#(U51(tt(),V2)) -> mark#(U52(isNatIListKind(V2)))
p47: mark#(U22(X1,X2)) -> active#(U22(mark(X1),X2))
p48: active#(U61(tt(),V2)) -> mark#(U62(isNatIListKind(V2)))
p49: mark#(U32(X1,X2)) -> active#(U32(mark(X1),X2))
p50: active#(U91(tt(),V1,V2)) -> mark#(U92(isNatKind(V1),V1,V2))
p51: mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3))
p52: active#(U92(tt(),V1,V2)) -> mark#(U93(isNatIListKind(V2),V1,V2))
p53: mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3))
p54: active#(U93(tt(),V1,V2)) -> mark#(U94(isNatIListKind(V2),V1,V2))
p55: mark#(U43(X1,X2,X3)) -> active#(U43(mark(X1),X2,X3))
p56: active#(U94(tt(),V1,V2)) -> mark#(U95(isNat(V1),V2))
p57: mark#(U44(X1,X2,X3)) -> active#(U44(mark(X1),X2,X3))
p58: active#(U95(tt(),V2)) -> mark#(U96(isNatList(V2)))
p59: mark#(U45(X1,X2)) -> active#(U45(mark(X1),X2))
p60: active#(isNat(s(V1))) -> mark#(U21(isNatKind(V1),V1))
p61: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p62: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNatKind(V1),V1,V2))
p63: mark#(U61(X1,X2)) -> active#(U61(mark(X1),X2))
p64: active#(isNatKind(s(V1))) -> mark#(U81(isNatKind(V1)))
p65: mark#(U91(X1,X2,X3)) -> active#(U91(mark(X1),X2,X3))
p66: active#(isNatList(cons(V1,V2))) -> mark#(U91(isNatKind(V1),V1,V2))
p67: mark#(U91(X1,X2,X3)) -> mark#(X1)
p68: mark#(U92(X1,X2,X3)) -> active#(U92(mark(X1),X2,X3))
p69: active#(isNatList(take(V1,V2))) -> mark#(U101(isNatKind(V1),V1,V2))
p70: mark#(U92(X1,X2,X3)) -> mark#(X1)
p71: mark#(U93(X1,X2,X3)) -> active#(U93(mark(X1),X2,X3))
p72: active#(length(cons(N,L))) -> mark#(U111(isNatList(L),L,N))
p73: mark#(U94(X1,X2,X3)) -> active#(U94(mark(X1),X2,X3))
p74: mark#(U95(X1,X2)) -> active#(U95(mark(X1),X2))
p75: mark#(U95(X1,X2)) -> mark#(X1)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41, r42, r43, r44, r45, r46, r47, r48, r49, r50, r51, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76, r77, r78, r79, r80, r81, r82, r83, r84, r85, r86, r87, r88, r89, r90, r91, r92, r93, r94, r95, r96, r97, r98, r99, r100, r101, r102, r103, r104, r105, r106, r107, r108, r109, r110, r111, r112, r113, r114, r115, r116, r117, r118, r119, r120, r121, r122, r123, r124, r125, r126, r127, r128, r129, r130, r131, r132, r133, r134, r135, r136, r137, r138, r139, r140, r141, r142, r143, r144, r145, r146, r147, r148, r149, r150, r151, r152, r153, r154, r155, r156, r157, r158, r159, r160, r161, r162, r163, r164, r165, r166, r167, r168, r169, r170, r171, r172, r173, r174, r175, r176, r177, r178, r179, r180, r181, r182, r183, r184, r185, r186, r187, r188, r189, r190, r191, r192, r193, r194, r195, r196, r197, r198, r199, r200, r201, r202, r203, r204, r205, r206, r207, r208, r209, r210, r211, r212, r213, r214, r215, r216, r217, r218, r219, r220, r221, r222, r223, r224, r225, r226, r227, r228, r229, r230, r231, r232, r233, r234, r235, r236, r237, r238, r239, r240, r241, r242, r243, r244, r245, r246, r247, r248, r249, r250, r251, r252, r253, r254, r255, r256, r257, r258, r259, r260, r261, r262, r263, r264, r265, r266, r267, r268, r269, r270, r271, r272, r273, r274, r275, r276, r277, r278, r279, r280, r281, r282, r283, r284, r285, r286, r287, r288, r289, r290, r291, r292, r293, r294, r295, r296, r297, r298, r299, r300, r301, r302, r303, r304, r305, r306, r307, r308, r309, r310, r311, r312, r313, r314, r315, r316, r317, r318, r319, r320, r321, r322, r323, r324, r325, r326, r327, r328, r329, r330, r331, r332, r333, r334, r335, r336, r337, r338, r339, r340, r341, r342, r343, r344, r345, r346, r347, r348, r349, r350, r351, r352, r353, r354, r355, r356, r357, r358

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    mark#_A(x1) = max{0, x1 - 237}
    U101_A(x1,x2,x3) = x1 + 673
    active#_A(x1) = max{0, x1 - 245}
    mark_A(x1) = x1 + 8
    tt_A = 56
    U102_A(x1,x2,x3) = max{648, x1 + 593}
    isNatKind_A(x1) = 128
    U104_A(x1,x2,x3) = 633
    U105_A(x1,x2) = x1 + 239
    isNat_A(x1) = 384
    U103_A(x1,x2,x3) = 641
    U106_A(x1) = 238
    isNatIList_A(x1) = max{1063, x1 - 9}
    isNatIListKind_A(x1) = x1 + 208
    U11_A(x1,x2) = 254
    U12_A(x1,x2) = 246
    U111_A(x1,x2,x3) = max{638, x1 + 584}
    U112_A(x1,x2,x3) = 631
    U113_A(x1,x2,x3) = max{246, x1 + 239}
    U114_A(x1,x2) = 245
    s_A(x1) = 79
    length_A(x1) = max{1412, x1 + 1404}
    U131_A(x1,x2,x3,x4) = 27
    U132_A(x1,x2,x3,x4) = 27
    U133_A(x1,x2,x3,x4) = 27
    U134_A(x1,x2,x3,x4) = 27
    U135_A(x1,x2,x3,x4) = 27
    U136_A(x1,x2,x3,x4) = max{27, x1 - 108}
    cons_A(x1,x2) = max{27, x2 - 3}
    take_A(x1,x2) = 30
    U21_A(x1,x2) = x1 + 247
    U22_A(x1,x2) = 238
    isNatList_A(x1) = x1 + 809
    U23_A(x1) = 217
    U32_A(x1,x2) = 1047
    U33_A(x1) = 238
    U41_A(x1,x2,x3) = max{x1 + 927, x3 - 21}
    U42_A(x1,x2,x3) = max{975, x1 - 49, x3 - 49}
    U43_A(x1,x2,x3) = max{967, x3 - 57}
    U44_A(x1,x2,x3) = max{255, x1 - 273, x3 - 217}
    U45_A(x1,x2) = max{246, x1 - 610, x2 - 225}
    U46_A(x1) = 238
    U51_A(x1,x2) = max{204, x1 - 494, x2 + 197}
    U52_A(x1) = max{188, x1 - 20}
    U61_A(x1,x2) = x1 + 56
    U62_A(x1) = 103
    U91_A(x1,x2,x3) = max{x1 + 108, x3 + 798}
    U92_A(x1,x2,x3) = max{x1 + 27, x3 + 642}
    U93_A(x1,x2,x3) = max{x1 + 172, x3 + 634}
    U94_A(x1,x2,x3) = max{421, x1 + 418, x3 + 419}
    U95_A(x1,x2) = max{418, x1 + 26, x2 + 411}
    U96_A(x1) = 73
    U81_A(x1) = 119
    active_A(x1) = max{35, x1}
    zeros_A = 30
    |0|_A = 28
    U13_A(x1) = 177
    U121_A(x1,x2) = 27
    U122_A(x1) = 27
    nil_A = 27
    U31_A(x1,x2) = 1055
    U71_A(x1) = 64

The next rules are strictly ordered:

  p4, p5, p6, p9, p10, p13, p14, p15, p17, p19, p23, p27, p30, p31, p34, p47, p49, p50, p55, p58, p60, p67, p69, p70, p75

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(U101(X1,X2,X3)) -> active#(U101(mark(X1),X2,X3))
p2: active#(U101(tt(),V1,V2)) -> mark#(U102(isNatKind(V1),V1,V2))
p3: mark#(isNatKind(X)) -> active#(isNatKind(X))
p4: mark#(isNatIListKind(X)) -> active#(isNatIListKind(X))
p5: active#(U11(tt(),V1)) -> mark#(U12(isNatIListKind(V1),V1))
p6: mark#(U105(X1,X2)) -> active#(U105(mark(X1),X2))
p7: active#(U112(tt(),L,N)) -> mark#(U113(isNat(N),L,N))
p8: active#(U114(tt(),L)) -> mark#(s(length(L)))
p9: active#(U131(tt(),IL,M,N)) -> mark#(U132(isNatIListKind(IL),IL,M,N))
p10: active#(U132(tt(),IL,M,N)) -> mark#(U133(isNat(M),IL,M,N))
p11: mark#(U111(X1,X2,X3)) -> active#(U111(mark(X1),X2,X3))
p12: active#(U133(tt(),IL,M,N)) -> mark#(U134(isNatKind(M),IL,M,N))
p13: active#(U134(tt(),IL,M,N)) -> mark#(U135(isNat(N),IL,M,N))
p14: mark#(U113(X1,X2,X3)) -> active#(U113(mark(X1),X2,X3))
p15: active#(U135(tt(),IL,M,N)) -> mark#(U136(isNatKind(N),IL,M,N))
p16: active#(U136(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL)))
p17: mark#(length(X)) -> active#(length(mark(X)))
p18: active#(U22(tt(),V1)) -> mark#(U23(isNat(V1)))
p19: mark#(U131(X1,X2,X3,X4)) -> active#(U131(mark(X1),X2,X3,X4))
p20: mark#(U132(X1,X2,X3,X4)) -> active#(U132(mark(X1),X2,X3,X4))
p21: active#(U41(tt(),V1,V2)) -> mark#(U42(isNatKind(V1),V1,V2))
p22: mark#(U133(X1,X2,X3,X4)) -> active#(U133(mark(X1),X2,X3,X4))
p23: active#(U42(tt(),V1,V2)) -> mark#(U43(isNatIListKind(V2),V1,V2))
p24: mark#(U134(X1,X2,X3,X4)) -> active#(U134(mark(X1),X2,X3,X4))
p25: active#(U43(tt(),V1,V2)) -> mark#(U44(isNatIListKind(V2),V1,V2))
p26: mark#(U135(X1,X2,X3,X4)) -> active#(U135(mark(X1),X2,X3,X4))
p27: active#(U44(tt(),V1,V2)) -> mark#(U45(isNat(V1),V2))
p28: mark#(U136(X1,X2,X3,X4)) -> active#(U136(mark(X1),X2,X3,X4))
p29: active#(U45(tt(),V2)) -> mark#(U46(isNatIList(V2)))
p30: mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2))
p31: active#(U51(tt(),V2)) -> mark#(U52(isNatIListKind(V2)))
p32: active#(U61(tt(),V2)) -> mark#(U62(isNatIListKind(V2)))
p33: mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3))
p34: active#(U92(tt(),V1,V2)) -> mark#(U93(isNatIListKind(V2),V1,V2))
p35: mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3))
p36: active#(U93(tt(),V1,V2)) -> mark#(U94(isNatIListKind(V2),V1,V2))
p37: active#(U94(tt(),V1,V2)) -> mark#(U95(isNat(V1),V2))
p38: mark#(U44(X1,X2,X3)) -> active#(U44(mark(X1),X2,X3))
p39: mark#(U45(X1,X2)) -> active#(U45(mark(X1),X2))
p40: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p41: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNatKind(V1),V1,V2))
p42: mark#(U61(X1,X2)) -> active#(U61(mark(X1),X2))
p43: active#(isNatKind(s(V1))) -> mark#(U81(isNatKind(V1)))
p44: mark#(U91(X1,X2,X3)) -> active#(U91(mark(X1),X2,X3))
p45: active#(isNatList(cons(V1,V2))) -> mark#(U91(isNatKind(V1),V1,V2))
p46: mark#(U92(X1,X2,X3)) -> active#(U92(mark(X1),X2,X3))
p47: mark#(U93(X1,X2,X3)) -> active#(U93(mark(X1),X2,X3))
p48: active#(length(cons(N,L))) -> mark#(U111(isNatList(L),L,N))
p49: mark#(U94(X1,X2,X3)) -> active#(U94(mark(X1),X2,X3))
p50: mark#(U95(X1,X2)) -> active#(U95(mark(X1),X2))

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19, p20, p21, p22, p23, p24, p25, p26, p27, p28, p29, p30, p31, p32, p33, p34, p35, p36, p37, p38, p39, p40, p41, p42, p43, p44, p45, p46, p47, p48, p49, p50}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(U101(X1,X2,X3)) -> active#(U101(mark(X1),X2,X3))
p2: active#(length(cons(N,L))) -> mark#(U111(isNatList(L),L,N))
p3: mark#(U95(X1,X2)) -> active#(U95(mark(X1),X2))
p4: active#(isNatList(cons(V1,V2))) -> mark#(U91(isNatKind(V1),V1,V2))
p5: mark#(U94(X1,X2,X3)) -> active#(U94(mark(X1),X2,X3))
p6: active#(isNatKind(s(V1))) -> mark#(U81(isNatKind(V1)))
p7: mark#(U93(X1,X2,X3)) -> active#(U93(mark(X1),X2,X3))
p8: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNatKind(V1),V1,V2))
p9: mark#(U92(X1,X2,X3)) -> active#(U92(mark(X1),X2,X3))
p10: active#(U94(tt(),V1,V2)) -> mark#(U95(isNat(V1),V2))
p11: mark#(U91(X1,X2,X3)) -> active#(U91(mark(X1),X2,X3))
p12: active#(U93(tt(),V1,V2)) -> mark#(U94(isNatIListKind(V2),V1,V2))
p13: mark#(U61(X1,X2)) -> active#(U61(mark(X1),X2))
p14: active#(U92(tt(),V1,V2)) -> mark#(U93(isNatIListKind(V2),V1,V2))
p15: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p16: active#(U61(tt(),V2)) -> mark#(U62(isNatIListKind(V2)))
p17: mark#(U45(X1,X2)) -> active#(U45(mark(X1),X2))
p18: active#(U51(tt(),V2)) -> mark#(U52(isNatIListKind(V2)))
p19: mark#(U44(X1,X2,X3)) -> active#(U44(mark(X1),X2,X3))
p20: active#(U45(tt(),V2)) -> mark#(U46(isNatIList(V2)))
p21: mark#(U42(X1,X2,X3)) -> active#(U42(mark(X1),X2,X3))
p22: active#(U44(tt(),V1,V2)) -> mark#(U45(isNat(V1),V2))
p23: mark#(U41(X1,X2,X3)) -> active#(U41(mark(X1),X2,X3))
p24: active#(U43(tt(),V1,V2)) -> mark#(U44(isNatIListKind(V2),V1,V2))
p25: mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2))
p26: active#(U42(tt(),V1,V2)) -> mark#(U43(isNatIListKind(V2),V1,V2))
p27: mark#(U136(X1,X2,X3,X4)) -> active#(U136(mark(X1),X2,X3,X4))
p28: active#(U41(tt(),V1,V2)) -> mark#(U42(isNatKind(V1),V1,V2))
p29: mark#(U135(X1,X2,X3,X4)) -> active#(U135(mark(X1),X2,X3,X4))
p30: active#(U22(tt(),V1)) -> mark#(U23(isNat(V1)))
p31: mark#(U134(X1,X2,X3,X4)) -> active#(U134(mark(X1),X2,X3,X4))
p32: active#(U136(tt(),IL,M,N)) -> mark#(cons(N,take(M,IL)))
p33: mark#(U133(X1,X2,X3,X4)) -> active#(U133(mark(X1),X2,X3,X4))
p34: active#(U135(tt(),IL,M,N)) -> mark#(U136(isNatKind(N),IL,M,N))
p35: mark#(U132(X1,X2,X3,X4)) -> active#(U132(mark(X1),X2,X3,X4))
p36: active#(U134(tt(),IL,M,N)) -> mark#(U135(isNat(N),IL,M,N))
p37: mark#(U131(X1,X2,X3,X4)) -> active#(U131(mark(X1),X2,X3,X4))
p38: active#(U133(tt(),IL,M,N)) -> mark#(U134(isNatKind(M),IL,M,N))
p39: mark#(length(X)) -> active#(length(mark(X)))
p40: active#(U132(tt(),IL,M,N)) -> mark#(U133(isNat(M),IL,M,N))
p41: mark#(U113(X1,X2,X3)) -> active#(U113(mark(X1),X2,X3))
p42: active#(U131(tt(),IL,M,N)) -> mark#(U132(isNatIListKind(IL),IL,M,N))
p43: mark#(U111(X1,X2,X3)) -> active#(U111(mark(X1),X2,X3))
p44: active#(U114(tt(),L)) -> mark#(s(length(L)))
p45: mark#(U105(X1,X2)) -> active#(U105(mark(X1),X2))
p46: active#(U112(tt(),L,N)) -> mark#(U113(isNat(N),L,N))
p47: mark#(isNatIListKind(X)) -> active#(isNatIListKind(X))
p48: active#(U11(tt(),V1)) -> mark#(U12(isNatIListKind(V1),V1))
p49: mark#(isNatKind(X)) -> active#(isNatKind(X))
p50: active#(U101(tt(),V1,V2)) -> mark#(U102(isNatKind(V1),V1,V2))

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41, r42, r43, r44, r45, r46, r47, r48, r49, r50, r51, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76, r77, r78, r79, r80, r81, r82, r83, r84, r85, r86, r87, r88, r89, r90, r91, r92, r93, r94, r95, r96, r97, r98, r99, r100, r101, r102, r103, r104, r105, r106, r107, r108, r109, r110, r111, r112, r113, r114, r115, r116, r117, r118, r119, r120, r121, r122, r123, r124, r125, r126, r127, r128, r129, r130, r131, r132, r133, r134, r135, r136, r137, r138, r139, r140, r141, r142, r143, r144, r145, r146, r147, r148, r149, r150, r151, r152, r153, r154, r155, r156, r157, r158, r159, r160, r161, r162, r163, r164, r165, r166, r167, r168, r169, r170, r171, r172, r173, r174, r175, r176, r177, r178, r179, r180, r181, r182, r183, r184, r185, r186, r187, r188, r189, r190, r191, r192, r193, r194, r195, r196, r197, r198, r199, r200, r201, r202, r203, r204, r205, r206, r207, r208, r209, r210, r211, r212, r213, r214, r215, r216, r217, r218, r219, r220, r221, r222, r223, r224, r225, r226, r227, r228, r229, r230, r231, r232, r233, r234, r235, r236, r237, r238, r239, r240, r241, r242, r243, r244, r245, r246, r247, r248, r249, r250, r251, r252, r253, r254, r255, r256, r257, r258, r259, r260, r261, r262, r263, r264, r265, r266, r267, r268, r269, r270, r271, r272, r273, r274, r275, r276, r277, r278, r279, r280, r281, r282, r283, r284, r285, r286, r287, r288, r289, r290, r291, r292, r293, r294, r295, r296, r297, r298, r299, r300, r301, r302, r303, r304, r305, r306, r307, r308, r309, r310, r311, r312, r313, r314, r315, r316, r317, r318, r319, r320, r321, r322, r323, r324, r325, r326, r327, r328, r329, r330, r331, r332, r333, r334, r335, r336, r337, r338, r339, r340, r341, r342, r343, r344, r345, r346, r347, r348, r349, r350, r351, r352, r353, r354, r355, r356, r357, r358

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    mark#_A(x1) = max{0, x1 - 13}
    U101_A(x1,x2,x3) = max{3940, x1 + 3891, x2 + 3841}
    active#_A(x1) = max{0, x1 - 67}
    mark_A(x1) = x1 + 49
    length_A(x1) = 342
    cons_A(x1,x2) = max{729, x2 - 2309}
    U111_A(x1,x2,x3) = 287
    isNatList_A(x1) = max{3989, x1 + 2377}
    U95_A(x1,x2) = max{11, x1 - 107}
    U91_A(x1,x2,x3) = max{13, x3 - 2880}
    isNatKind_A(x1) = x1 + 649
    U94_A(x1,x2,x3) = 11
    s_A(x1) = max{80, x1 - 330}
    U81_A(x1) = 270
    U93_A(x1,x2,x3) = 11
    isNatIList_A(x1) = x1 + 2998
    U41_A(x1,x2,x3) = x3 + 635
    U92_A(x1,x2,x3) = 11
    tt_A = 11
    isNat_A(x1) = 118
    isNatIListKind_A(x1) = 11
    U61_A(x1,x2) = 11
    U51_A(x1,x2) = 11
    U62_A(x1) = 11
    U45_A(x1,x2) = max{411, x1 + 62, x2 + 10}
    U52_A(x1) = 11
    U44_A(x1,x2,x3) = max{465, x1 + 416, x3 + 64}
    U46_A(x1) = 357
    U42_A(x1,x2,x3) = x3 + 581
    U43_A(x1,x2,x3) = max{x1 + 508, x3 + 482}
    U21_A(x1,x2) = 12
    U136_A(x1,x2,x3,x4) = max{817, x3 - 42}
    U135_A(x1,x2,x3,x4) = max{872, x1 + 57, x3 + 13}
    U22_A(x1,x2) = 11
    U23_A(x1) = max{11, x1 - 162}
    U134_A(x1,x2,x3,x4) = max{926, x1 + 219, x3 + 67}
    take_A(x1,x2) = x1 + 2212
    U133_A(x1,x2,x3,x4) = max{x1 + 969, x3 + 923}
    U132_A(x1,x2,x3,x4) = max{x1 + 1130, x3 + 1141}
    U131_A(x1,x2,x3,x4) = x3 + 1195
    U113_A(x1,x2,x3) = max{184, x1 + 64}
    U114_A(x1,x2) = 134
    U105_A(x1,x2) = x1 + 159
    U112_A(x1,x2,x3) = max{238, x1 + 226}
    U11_A(x1,x2) = max{68, x1 + 58}
    U12_A(x1,x2) = 15
    U102_A(x1,x2,x3) = x1 + 3138
    active_A(x1) = max{60, x1}
    zeros_A = 2321
    |0|_A = 293
    U103_A(x1,x2,x3) = max{375, x1 + 364}
    U104_A(x1,x2,x3) = max{326, x1 + 315}
    U106_A(x1) = 60
    U13_A(x1) = 11
    U121_A(x1,x2) = 110
    U122_A(x1) = 61
    nil_A = 12
    U31_A(x1,x2) = 157
    U32_A(x1,x2) = 108
    U33_A(x1) = 59
    U71_A(x1) = max{60, x1 + 49}
    U96_A(x1) = 11

The next rules are strictly ordered:

  p1, p2, p4, p6, p17, p19, p21, p23, p26, p27, p29, p31, p32, p33, p34, p35, p37, p39, p41, p43, p45, p49

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(U95(X1,X2)) -> active#(U95(mark(X1),X2))
p2: mark#(U94(X1,X2,X3)) -> active#(U94(mark(X1),X2,X3))
p3: mark#(U93(X1,X2,X3)) -> active#(U93(mark(X1),X2,X3))
p4: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNatKind(V1),V1,V2))
p5: mark#(U92(X1,X2,X3)) -> active#(U92(mark(X1),X2,X3))
p6: active#(U94(tt(),V1,V2)) -> mark#(U95(isNat(V1),V2))
p7: mark#(U91(X1,X2,X3)) -> active#(U91(mark(X1),X2,X3))
p8: active#(U93(tt(),V1,V2)) -> mark#(U94(isNatIListKind(V2),V1,V2))
p9: mark#(U61(X1,X2)) -> active#(U61(mark(X1),X2))
p10: active#(U92(tt(),V1,V2)) -> mark#(U93(isNatIListKind(V2),V1,V2))
p11: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p12: active#(U61(tt(),V2)) -> mark#(U62(isNatIListKind(V2)))
p13: active#(U51(tt(),V2)) -> mark#(U52(isNatIListKind(V2)))
p14: active#(U45(tt(),V2)) -> mark#(U46(isNatIList(V2)))
p15: active#(U44(tt(),V1,V2)) -> mark#(U45(isNat(V1),V2))
p16: active#(U43(tt(),V1,V2)) -> mark#(U44(isNatIListKind(V2),V1,V2))
p17: mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2))
p18: active#(U41(tt(),V1,V2)) -> mark#(U42(isNatKind(V1),V1,V2))
p19: active#(U22(tt(),V1)) -> mark#(U23(isNat(V1)))
p20: active#(U134(tt(),IL,M,N)) -> mark#(U135(isNat(N),IL,M,N))
p21: active#(U133(tt(),IL,M,N)) -> mark#(U134(isNatKind(M),IL,M,N))
p22: active#(U132(tt(),IL,M,N)) -> mark#(U133(isNat(M),IL,M,N))
p23: active#(U131(tt(),IL,M,N)) -> mark#(U132(isNatIListKind(IL),IL,M,N))
p24: active#(U114(tt(),L)) -> mark#(s(length(L)))
p25: active#(U112(tt(),L,N)) -> mark#(U113(isNat(N),L,N))
p26: mark#(isNatIListKind(X)) -> active#(isNatIListKind(X))
p27: active#(U11(tt(),V1)) -> mark#(U12(isNatIListKind(V1),V1))
p28: active#(U101(tt(),V1,V2)) -> mark#(U102(isNatKind(V1),V1,V2))

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19, p20, p21, p22, p23, p24, p25, p26, p27, p28}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(U95(X1,X2)) -> active#(U95(mark(X1),X2))
p2: active#(U101(tt(),V1,V2)) -> mark#(U102(isNatKind(V1),V1,V2))
p3: mark#(isNatIListKind(X)) -> active#(isNatIListKind(X))
p4: active#(U11(tt(),V1)) -> mark#(U12(isNatIListKind(V1),V1))
p5: mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2))
p6: active#(U112(tt(),L,N)) -> mark#(U113(isNat(N),L,N))
p7: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p8: active#(U114(tt(),L)) -> mark#(s(length(L)))
p9: mark#(U61(X1,X2)) -> active#(U61(mark(X1),X2))
p10: active#(U131(tt(),IL,M,N)) -> mark#(U132(isNatIListKind(IL),IL,M,N))
p11: mark#(U91(X1,X2,X3)) -> active#(U91(mark(X1),X2,X3))
p12: active#(U132(tt(),IL,M,N)) -> mark#(U133(isNat(M),IL,M,N))
p13: mark#(U92(X1,X2,X3)) -> active#(U92(mark(X1),X2,X3))
p14: active#(U133(tt(),IL,M,N)) -> mark#(U134(isNatKind(M),IL,M,N))
p15: mark#(U93(X1,X2,X3)) -> active#(U93(mark(X1),X2,X3))
p16: active#(U134(tt(),IL,M,N)) -> mark#(U135(isNat(N),IL,M,N))
p17: mark#(U94(X1,X2,X3)) -> active#(U94(mark(X1),X2,X3))
p18: active#(U22(tt(),V1)) -> mark#(U23(isNat(V1)))
p19: active#(U41(tt(),V1,V2)) -> mark#(U42(isNatKind(V1),V1,V2))
p20: active#(U43(tt(),V1,V2)) -> mark#(U44(isNatIListKind(V2),V1,V2))
p21: active#(U44(tt(),V1,V2)) -> mark#(U45(isNat(V1),V2))
p22: active#(U45(tt(),V2)) -> mark#(U46(isNatIList(V2)))
p23: active#(U51(tt(),V2)) -> mark#(U52(isNatIListKind(V2)))
p24: active#(U61(tt(),V2)) -> mark#(U62(isNatIListKind(V2)))
p25: active#(U92(tt(),V1,V2)) -> mark#(U93(isNatIListKind(V2),V1,V2))
p26: active#(U93(tt(),V1,V2)) -> mark#(U94(isNatIListKind(V2),V1,V2))
p27: active#(U94(tt(),V1,V2)) -> mark#(U95(isNat(V1),V2))
p28: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNatKind(V1),V1,V2))

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41, r42, r43, r44, r45, r46, r47, r48, r49, r50, r51, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76, r77, r78, r79, r80, r81, r82, r83, r84, r85, r86, r87, r88, r89, r90, r91, r92, r93, r94, r95, r96, r97, r98, r99, r100, r101, r102, r103, r104, r105, r106, r107, r108, r109, r110, r111, r112, r113, r114, r115, r116, r117, r118, r119, r120, r121, r122, r123, r124, r125, r126, r127, r128, r129, r130, r131, r132, r133, r134, r135, r136, r137, r138, r139, r140, r141, r142, r143, r144, r145, r146, r147, r148, r149, r150, r151, r152, r153, r154, r155, r156, r157, r158, r159, r160, r161, r162, r163, r164, r165, r166, r167, r168, r169, r170, r171, r172, r173, r174, r175, r176, r177, r178, r179, r180, r181, r182, r183, r184, r185, r186, r187, r188, r189, r190, r191, r192, r193, r194, r195, r196, r197, r198, r199, r200, r201, r202, r203, r204, r205, r206, r207, r208, r209, r210, r211, r212, r213, r214, r215, r216, r217, r218, r219, r220, r221, r222, r223, r224, r225, r226, r227, r228, r229, r230, r231, r232, r233, r234, r235, r236, r237, r238, r239, r240, r241, r242, r243, r244, r245, r246, r247, r248, r249, r250, r251, r252, r253, r254, r255, r256, r257, r258, r259, r260, r261, r262, r263, r264, r265, r266, r267, r268, r269, r270, r271, r272, r273, r274, r275, r276, r277, r278, r279, r280, r281, r282, r283, r284, r285, r286, r287, r288, r289, r290, r291, r292, r293, r294, r295, r296, r297, r298, r299, r300, r301, r302, r303, r304, r305, r306, r307, r308, r309, r310, r311, r312, r313, r314, r315, r316, r317, r318, r319, r320, r321, r322, r323, r324, r325, r326, r327, r328, r329, r330, r331, r332, r333, r334, r335, r336, r337, r338, r339, r340, r341, r342, r343, r344, r345, r346, r347, r348, r349, r350, r351, r352, r353, r354, r355, r356, r357, r358

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    mark#_A(x1) = max{0, x1 - 1279}
    U95_A(x1,x2) = 1232
    active#_A(x1) = max{0, x1 - 1344}
    mark_A(x1) = max{63, x1 + 47}
    U101_A(x1,x2,x3) = max{3609, x1 + 3466}
    tt_A = 103
    U102_A(x1,x2,x3) = max{3543, x1 - 3908}
    isNatKind_A(x1) = 143
    isNatIListKind_A(x1) = x1 + 1279
    U11_A(x1,x2) = 103
    U12_A(x1,x2) = 103
    U21_A(x1,x2) = 103
    U112_A(x1,x2,x3) = max{x1 + 102, x2 + 1345}
    U113_A(x1,x2,x3) = max{x1 + 49, x2 + 1280}
    isNat_A(x1) = 145
    U51_A(x1,x2) = 103
    U114_A(x1,x2) = 149
    s_A(x1) = 103
    length_A(x1) = max{2755, x1 + 2739}
    U61_A(x1,x2) = 103
    U131_A(x1,x2,x3,x4) = 1540
    U132_A(x1,x2,x3,x4) = 1474
    U91_A(x1,x2,x3) = max{x1 - 2639, x3 + 1488}
    U133_A(x1,x2,x3,x4) = max{1409, x1 + 1263}
    U92_A(x1,x2,x3) = max{x1 + 1298, x3 + 1401}
    U134_A(x1,x2,x3,x4) = 1343
    U93_A(x1,x2,x3) = 1336
    U135_A(x1,x2,x3,x4) = 1147
    U94_A(x1,x2,x3) = 1279
    U22_A(x1,x2) = 103
    U23_A(x1) = 103
    U41_A(x1,x2,x3) = max{292, x1 + 60, x3 + 274}
    U42_A(x1,x2,x3) = max{x1 + 102, x3 + 204}
    U43_A(x1,x2,x3) = max{158, x3 + 157}
    U44_A(x1,x2,x3) = max{110, x1 - 1188}
    U45_A(x1,x2) = max{103, x1 - 1277}
    U46_A(x1) = 103
    isNatIList_A(x1) = max{1344, x1 + 873}
    U52_A(x1) = 103
    U62_A(x1) = 103
    cons_A(x1,x2) = max{471, x2 - 534}
    active_A(x1) = max{150, x1}
    zeros_A = 638
    |0|_A = 4052
    U103_A(x1,x2,x3) = 106
    U104_A(x1,x2,x3) = 103
    U105_A(x1,x2) = 103
    U106_A(x1) = 103
    U111_A(x1,x2,x3) = max{x1 + 89, x2 + 1428}
    U13_A(x1) = 103
    isNatList_A(x1) = x1 + 2069
    U121_A(x1,x2) = 1540
    U122_A(x1) = 1407
    nil_A = 1360
    U136_A(x1,x2,x3,x4) = max{1100, x1 + 102}
    take_A(x1,x2) = 1587
    U31_A(x1,x2) = max{x1 - 453, x2 + 826}
    U32_A(x1,x2) = x2 + 197
    U33_A(x1) = 150
    U71_A(x1) = 103
    U81_A(x1) = max{103, x1 - 40}
    U96_A(x1) = 104

The next rules are strictly ordered:

  p2, p10, p11, p13, p14, p15

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(U95(X1,X2)) -> active#(U95(mark(X1),X2))
p2: mark#(isNatIListKind(X)) -> active#(isNatIListKind(X))
p3: active#(U11(tt(),V1)) -> mark#(U12(isNatIListKind(V1),V1))
p4: mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2))
p5: active#(U112(tt(),L,N)) -> mark#(U113(isNat(N),L,N))
p6: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p7: active#(U114(tt(),L)) -> mark#(s(length(L)))
p8: mark#(U61(X1,X2)) -> active#(U61(mark(X1),X2))
p9: active#(U132(tt(),IL,M,N)) -> mark#(U133(isNat(M),IL,M,N))
p10: active#(U134(tt(),IL,M,N)) -> mark#(U135(isNat(N),IL,M,N))
p11: mark#(U94(X1,X2,X3)) -> active#(U94(mark(X1),X2,X3))
p12: active#(U22(tt(),V1)) -> mark#(U23(isNat(V1)))
p13: active#(U41(tt(),V1,V2)) -> mark#(U42(isNatKind(V1),V1,V2))
p14: active#(U43(tt(),V1,V2)) -> mark#(U44(isNatIListKind(V2),V1,V2))
p15: active#(U44(tt(),V1,V2)) -> mark#(U45(isNat(V1),V2))
p16: active#(U45(tt(),V2)) -> mark#(U46(isNatIList(V2)))
p17: active#(U51(tt(),V2)) -> mark#(U52(isNatIListKind(V2)))
p18: active#(U61(tt(),V2)) -> mark#(U62(isNatIListKind(V2)))
p19: active#(U92(tt(),V1,V2)) -> mark#(U93(isNatIListKind(V2),V1,V2))
p20: active#(U93(tt(),V1,V2)) -> mark#(U94(isNatIListKind(V2),V1,V2))
p21: active#(U94(tt(),V1,V2)) -> mark#(U95(isNat(V1),V2))
p22: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNatKind(V1),V1,V2))

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7, p8, p9, p10, p11, p12, p13, p14, p15, p16, p17, p18, p19, p20, p21, p22}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: mark#(U95(X1,X2)) -> active#(U95(mark(X1),X2))
p2: active#(isNatIList(cons(V1,V2))) -> mark#(U41(isNatKind(V1),V1,V2))
p3: mark#(U94(X1,X2,X3)) -> active#(U94(mark(X1),X2,X3))
p4: active#(U94(tt(),V1,V2)) -> mark#(U95(isNat(V1),V2))
p5: mark#(U61(X1,X2)) -> active#(U61(mark(X1),X2))
p6: active#(U93(tt(),V1,V2)) -> mark#(U94(isNatIListKind(V2),V1,V2))
p7: mark#(U51(X1,X2)) -> active#(U51(mark(X1),X2))
p8: active#(U92(tt(),V1,V2)) -> mark#(U93(isNatIListKind(V2),V1,V2))
p9: mark#(U21(X1,X2)) -> active#(U21(mark(X1),X2))
p10: active#(U61(tt(),V2)) -> mark#(U62(isNatIListKind(V2)))
p11: mark#(isNatIListKind(X)) -> active#(isNatIListKind(X))
p12: active#(U51(tt(),V2)) -> mark#(U52(isNatIListKind(V2)))
p13: active#(U45(tt(),V2)) -> mark#(U46(isNatIList(V2)))
p14: active#(U44(tt(),V1,V2)) -> mark#(U45(isNat(V1),V2))
p15: active#(U43(tt(),V1,V2)) -> mark#(U44(isNatIListKind(V2),V1,V2))
p16: active#(U41(tt(),V1,V2)) -> mark#(U42(isNatKind(V1),V1,V2))
p17: active#(U22(tt(),V1)) -> mark#(U23(isNat(V1)))
p18: active#(U134(tt(),IL,M,N)) -> mark#(U135(isNat(N),IL,M,N))
p19: active#(U132(tt(),IL,M,N)) -> mark#(U133(isNat(M),IL,M,N))
p20: active#(U114(tt(),L)) -> mark#(s(length(L)))
p21: active#(U112(tt(),L,N)) -> mark#(U113(isNat(N),L,N))
p22: active#(U11(tt(),V1)) -> mark#(U12(isNatIListKind(V1),V1))

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, r12, r13, r14, r15, r16, r17, r18, r19, r20, r21, r22, r23, r24, r25, r26, r27, r28, r29, r30, r31, r32, r33, r34, r35, r36, r37, r38, r39, r40, r41, r42, r43, r44, r45, r46, r47, r48, r49, r50, r51, r52, r53, r54, r55, r56, r57, r58, r59, r60, r61, r62, r63, r64, r65, r66, r67, r68, r69, r70, r71, r72, r73, r74, r75, r76, r77, r78, r79, r80, r81, r82, r83, r84, r85, r86, r87, r88, r89, r90, r91, r92, r93, r94, r95, r96, r97, r98, r99, r100, r101, r102, r103, r104, r105, r106, r107, r108, r109, r110, r111, r112, r113, r114, r115, r116, r117, r118, r119, r120, r121, r122, r123, r124, r125, r126, r127, r128, r129, r130, r131, r132, r133, r134, r135, r136, r137, r138, r139, r140, r141, r142, r143, r144, r145, r146, r147, r148, r149, r150, r151, r152, r153, r154, r155, r156, r157, r158, r159, r160, r161, r162, r163, r164, r165, r166, r167, r168, r169, r170, r171, r172, r173, r174, r175, r176, r177, r178, r179, r180, r181, r182, r183, r184, r185, r186, r187, r188, r189, r190, r191, r192, r193, r194, r195, r196, r197, r198, r199, r200, r201, r202, r203, r204, r205, r206, r207, r208, r209, r210, r211, r212, r213, r214, r215, r216, r217, r218, r219, r220, r221, r222, r223, r224, r225, r226, r227, r228, r229, r230, r231, r232, r233, r234, r235, r236, r237, r238, r239, r240, r241, r242, r243, r244, r245, r246, r247, r248, r249, r250, r251, r252, r253, r254, r255, r256, r257, r258, r259, r260, r261, r262, r263, r264, r265, r266, r267, r268, r269, r270, r271, r272, r273, r274, r275, r276, r277, r278, r279, r280, r281, r282, r283, r284, r285, r286, r287, r288, r289, r290, r291, r292, r293, r294, r295, r296, r297, r298, r299, r300, r301, r302, r303, r304, r305, r306, r307, r308, r309, r310, r311, r312, r313, r314, r315, r316, r317, r318, r319, r320, r321, r322, r323, r324, r325, r326, r327, r328, r329, r330, r331, r332, r333, r334, r335, r336, r337, r338, r339, r340, r341, r342, r343, r344, r345, r346, r347, r348, r349, r350, r351, r352, r353, r354, r355, r356, r357, r358

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    mark#_A(x1) = x1 + 4559
    U95_A(x1,x2) = x2 + 318
    active#_A(x1) = max{4564, x1 + 4535}
    mark_A(x1) = max{46, x1 + 39}
    isNatIList_A(x1) = max{1500, x1 + 1460}
    cons_A(x1,x2) = max{7, x2 - 244}
    U41_A(x1,x2,x3) = max{1178, x1 + 1087, x3 + 1078}
    isNatKind_A(x1) = 90
    U94_A(x1,x2,x3) = x3 + 359
    tt_A = 7
    isNat_A(x1) = 17
    U61_A(x1,x2) = 6
    U93_A(x1,x2,x3) = max{x1 - 33, x3 + 659}
    isNatIListKind_A(x1) = x1 + 314
    U51_A(x1,x2) = x2 + 31
    U92_A(x1,x2,x3) = max{x1 + 26, x3 + 698}
    U21_A(x1,x2) = 6
    U62_A(x1) = 5
    U52_A(x1) = 7
    U45_A(x1,x2) = max{7, x1 - 9, x2 - 2}
    U46_A(x1) = 5
    U44_A(x1,x2,x3) = max{77, x1 + 38, x3 + 37}
    U43_A(x1,x2,x3) = max{x1 + 108, x3 + 391}
    U42_A(x1,x2,x3) = max{1055, x3 + 1015}
    U22_A(x1,x2) = max{7, x1 - 83}
    U23_A(x1) = 4
    U134_A(x1,x2,x3,x4) = max{7, x1 - 3803}
    U135_A(x1,x2,x3,x4) = max{0, x1 - 4999}
    U132_A(x1,x2,x3,x4) = 30
    U133_A(x1,x2,x3,x4) = max{6, x1 - 3681}
    U114_A(x1,x2) = max{x1 + 1130, x2 + 1137}
    s_A(x1) = max{1098, x1 - 2585}
    length_A(x1) = max{2605, x1 + 1620}
    U112_A(x1,x2,x3) = max{x1 + 126, x2 + 1298}
    U113_A(x1,x2,x3) = max{x1 + 77, x2 + 1259}
    U11_A(x1,x2) = 7
    U12_A(x1,x2) = 5
    active_A(x1) = max{46, x1}
    zeros_A = 0
    |0|_A = 2566
    U101_A(x1,x2,x3) = max{714, x1 + 675}
    U102_A(x1,x2,x3) = max{624, x1 + 585}
    U103_A(x1,x2,x3) = 45
    U104_A(x1,x2,x3) = 7
    U105_A(x1,x2) = 6
    U106_A(x1) = 6
    U111_A(x1,x2,x3) = max{x1 + 151, x2 + 1337}
    U13_A(x1) = 7
    isNatList_A(x1) = x1 + 1020
    U121_A(x1,x2) = 90
    U122_A(x1) = 47
    nil_A = 0
    U131_A(x1,x2,x3,x4) = 83
    U136_A(x1,x2,x3,x4) = max{6, x1 - 83}
    take_A(x1,x2) = 129
    U31_A(x1,x2) = max{1146, x1 + 1107, x2 + 1106}
    U32_A(x1,x2) = max{x1 + 317, x2 + 1067}
    U33_A(x1) = max{47, x1 + 8}
    U71_A(x1) = 47
    U81_A(x1) = 51
    U91_A(x1,x2,x3) = max{x1 + 148, x3 + 737}
    U96_A(x1) = 8

The next rules are strictly ordered:

  p1, p2, p3, p4, p5, p6, p7, p8, p9, p11, p14, p15, p16, p17, p18, p20, p21

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: active#(U61(tt(),V2)) -> mark#(U62(isNatIListKind(V2)))
p2: active#(U51(tt(),V2)) -> mark#(U52(isNatIListKind(V2)))
p3: active#(U45(tt(),V2)) -> mark#(U46(isNatIList(V2)))
p4: active#(U132(tt(),IL,M,N)) -> mark#(U133(isNat(M),IL,M,N))
p5: active#(U11(tt(),V1)) -> mark#(U12(isNatIListKind(V1),V1))

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  (no SCCs)

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U102#(mark(X1),X2,X3) -> U102#(X1,X2,X3)
p2: U102#(X1,X2,active(X3)) -> U102#(X1,X2,X3)
p3: U102#(X1,active(X2),X3) -> U102#(X1,X2,X3)
p4: U102#(active(X1),X2,X3) -> U102#(X1,X2,X3)
p5: U102#(X1,X2,mark(X3)) -> U102#(X1,X2,X3)
p6: U102#(X1,mark(X2),X3) -> U102#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U102#_A(x1,x2,x3) = x3 + 1
    mark_A(x1) = x1 + 1
    active_A(x1) = x1

The next rules are strictly ordered:

  p5

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U102#(mark(X1),X2,X3) -> U102#(X1,X2,X3)
p2: U102#(X1,X2,active(X3)) -> U102#(X1,X2,X3)
p3: U102#(X1,active(X2),X3) -> U102#(X1,X2,X3)
p4: U102#(active(X1),X2,X3) -> U102#(X1,X2,X3)
p5: U102#(X1,mark(X2),X3) -> U102#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U102#(mark(X1),X2,X3) -> U102#(X1,X2,X3)
p2: U102#(X1,mark(X2),X3) -> U102#(X1,X2,X3)
p3: U102#(active(X1),X2,X3) -> U102#(X1,X2,X3)
p4: U102#(X1,active(X2),X3) -> U102#(X1,X2,X3)
p5: U102#(X1,X2,active(X3)) -> U102#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U102#_A(x1,x2,x3) = x3
    mark_A(x1) = x1 + 2
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p5

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U102#(mark(X1),X2,X3) -> U102#(X1,X2,X3)
p2: U102#(X1,mark(X2),X3) -> U102#(X1,X2,X3)
p3: U102#(active(X1),X2,X3) -> U102#(X1,X2,X3)
p4: U102#(X1,active(X2),X3) -> U102#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U102#(mark(X1),X2,X3) -> U102#(X1,X2,X3)
p2: U102#(X1,active(X2),X3) -> U102#(X1,X2,X3)
p3: U102#(active(X1),X2,X3) -> U102#(X1,X2,X3)
p4: U102#(X1,mark(X2),X3) -> U102#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U102#_A(x1,x2,x3) = max{0, x2 - 1}
    mark_A(x1) = x1 + 2
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U102#(mark(X1),X2,X3) -> U102#(X1,X2,X3)
p2: U102#(X1,active(X2),X3) -> U102#(X1,X2,X3)
p3: U102#(active(X1),X2,X3) -> U102#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U102#(mark(X1),X2,X3) -> U102#(X1,X2,X3)
p2: U102#(active(X1),X2,X3) -> U102#(X1,X2,X3)
p3: U102#(X1,active(X2),X3) -> U102#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U102#_A(x1,x2,x3) = x1 + 1
    mark_A(x1) = x1
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U102#(mark(X1),X2,X3) -> U102#(X1,X2,X3)
p2: U102#(X1,active(X2),X3) -> U102#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U102#(mark(X1),X2,X3) -> U102#(X1,X2,X3)
p2: U102#(X1,active(X2),X3) -> U102#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U102#_A(x1,x2,x3) = x1
    mark_A(x1) = x1 + 1
    active_A(x1) = max{0, x1 - 1}

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U102#(X1,active(X2),X3) -> U102#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U102#(X1,active(X2),X3) -> U102#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U102#_A(x1,x2,x3) = x2
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: isNatKind#(mark(X)) -> isNatKind#(X)
p2: isNatKind#(active(X)) -> isNatKind#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    isNatKind#_A(x1) = max{0, x1 - 1}
    mark_A(x1) = x1 + 1
    active_A(x1) = x1 + 2

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: isNatKind#(mark(X)) -> isNatKind#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: isNatKind#(mark(X)) -> isNatKind#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  max/plus interpretations on natural numbers:
  
    isNatKind#_A(x1) = x1
    mark_A(x1) = x1 + 1

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U103#(mark(X1),X2,X3) -> U103#(X1,X2,X3)
p2: U103#(X1,X2,active(X3)) -> U103#(X1,X2,X3)
p3: U103#(X1,active(X2),X3) -> U103#(X1,X2,X3)
p4: U103#(active(X1),X2,X3) -> U103#(X1,X2,X3)
p5: U103#(X1,X2,mark(X3)) -> U103#(X1,X2,X3)
p6: U103#(X1,mark(X2),X3) -> U103#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U103#_A(x1,x2,x3) = x3 + 1
    mark_A(x1) = x1 + 1
    active_A(x1) = x1

The next rules are strictly ordered:

  p5

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U103#(mark(X1),X2,X3) -> U103#(X1,X2,X3)
p2: U103#(X1,X2,active(X3)) -> U103#(X1,X2,X3)
p3: U103#(X1,active(X2),X3) -> U103#(X1,X2,X3)
p4: U103#(active(X1),X2,X3) -> U103#(X1,X2,X3)
p5: U103#(X1,mark(X2),X3) -> U103#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U103#(mark(X1),X2,X3) -> U103#(X1,X2,X3)
p2: U103#(X1,mark(X2),X3) -> U103#(X1,X2,X3)
p3: U103#(active(X1),X2,X3) -> U103#(X1,X2,X3)
p4: U103#(X1,active(X2),X3) -> U103#(X1,X2,X3)
p5: U103#(X1,X2,active(X3)) -> U103#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U103#_A(x1,x2,x3) = x3
    mark_A(x1) = x1 + 2
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p5

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U103#(mark(X1),X2,X3) -> U103#(X1,X2,X3)
p2: U103#(X1,mark(X2),X3) -> U103#(X1,X2,X3)
p3: U103#(active(X1),X2,X3) -> U103#(X1,X2,X3)
p4: U103#(X1,active(X2),X3) -> U103#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U103#(mark(X1),X2,X3) -> U103#(X1,X2,X3)
p2: U103#(X1,active(X2),X3) -> U103#(X1,X2,X3)
p3: U103#(active(X1),X2,X3) -> U103#(X1,X2,X3)
p4: U103#(X1,mark(X2),X3) -> U103#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U103#_A(x1,x2,x3) = max{2, x2 + 1}
    mark_A(x1) = x1 + 2
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U103#(mark(X1),X2,X3) -> U103#(X1,X2,X3)
p2: U103#(X1,active(X2),X3) -> U103#(X1,X2,X3)
p3: U103#(active(X1),X2,X3) -> U103#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U103#(mark(X1),X2,X3) -> U103#(X1,X2,X3)
p2: U103#(active(X1),X2,X3) -> U103#(X1,X2,X3)
p3: U103#(X1,active(X2),X3) -> U103#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U103#_A(x1,x2,x3) = x1 + 1
    mark_A(x1) = x1
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U103#(mark(X1),X2,X3) -> U103#(X1,X2,X3)
p2: U103#(X1,active(X2),X3) -> U103#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U103#(mark(X1),X2,X3) -> U103#(X1,X2,X3)
p2: U103#(X1,active(X2),X3) -> U103#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U103#_A(x1,x2,x3) = x1
    mark_A(x1) = x1 + 1
    active_A(x1) = max{0, x1 - 1}

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U103#(X1,active(X2),X3) -> U103#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U103#(X1,active(X2),X3) -> U103#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U103#_A(x1,x2,x3) = x2
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: isNatIListKind#(mark(X)) -> isNatIListKind#(X)
p2: isNatIListKind#(active(X)) -> isNatIListKind#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    isNatIListKind#_A(x1) = max{0, x1 - 1}
    mark_A(x1) = x1 + 1
    active_A(x1) = x1 + 2

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: isNatIListKind#(mark(X)) -> isNatIListKind#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: isNatIListKind#(mark(X)) -> isNatIListKind#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  max/plus interpretations on natural numbers:
  
    isNatIListKind#_A(x1) = x1
    mark_A(x1) = x1 + 1

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U104#(mark(X1),X2,X3) -> U104#(X1,X2,X3)
p2: U104#(X1,X2,active(X3)) -> U104#(X1,X2,X3)
p3: U104#(X1,active(X2),X3) -> U104#(X1,X2,X3)
p4: U104#(active(X1),X2,X3) -> U104#(X1,X2,X3)
p5: U104#(X1,X2,mark(X3)) -> U104#(X1,X2,X3)
p6: U104#(X1,mark(X2),X3) -> U104#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U104#_A(x1,x2,x3) = x3 + 1
    mark_A(x1) = x1 + 1
    active_A(x1) = x1

The next rules are strictly ordered:

  p5

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U104#(mark(X1),X2,X3) -> U104#(X1,X2,X3)
p2: U104#(X1,X2,active(X3)) -> U104#(X1,X2,X3)
p3: U104#(X1,active(X2),X3) -> U104#(X1,X2,X3)
p4: U104#(active(X1),X2,X3) -> U104#(X1,X2,X3)
p5: U104#(X1,mark(X2),X3) -> U104#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U104#(mark(X1),X2,X3) -> U104#(X1,X2,X3)
p2: U104#(X1,mark(X2),X3) -> U104#(X1,X2,X3)
p3: U104#(active(X1),X2,X3) -> U104#(X1,X2,X3)
p4: U104#(X1,active(X2),X3) -> U104#(X1,X2,X3)
p5: U104#(X1,X2,active(X3)) -> U104#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U104#_A(x1,x2,x3) = x3
    mark_A(x1) = x1 + 2
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p5

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U104#(mark(X1),X2,X3) -> U104#(X1,X2,X3)
p2: U104#(X1,mark(X2),X3) -> U104#(X1,X2,X3)
p3: U104#(active(X1),X2,X3) -> U104#(X1,X2,X3)
p4: U104#(X1,active(X2),X3) -> U104#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U104#(mark(X1),X2,X3) -> U104#(X1,X2,X3)
p2: U104#(X1,active(X2),X3) -> U104#(X1,X2,X3)
p3: U104#(active(X1),X2,X3) -> U104#(X1,X2,X3)
p4: U104#(X1,mark(X2),X3) -> U104#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U104#_A(x1,x2,x3) = max{0, x2 - 1}
    mark_A(x1) = x1 + 2
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U104#(mark(X1),X2,X3) -> U104#(X1,X2,X3)
p2: U104#(X1,active(X2),X3) -> U104#(X1,X2,X3)
p3: U104#(active(X1),X2,X3) -> U104#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U104#(mark(X1),X2,X3) -> U104#(X1,X2,X3)
p2: U104#(active(X1),X2,X3) -> U104#(X1,X2,X3)
p3: U104#(X1,active(X2),X3) -> U104#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U104#_A(x1,x2,x3) = x1 + 1
    mark_A(x1) = x1
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U104#(mark(X1),X2,X3) -> U104#(X1,X2,X3)
p2: U104#(X1,active(X2),X3) -> U104#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U104#(mark(X1),X2,X3) -> U104#(X1,X2,X3)
p2: U104#(X1,active(X2),X3) -> U104#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U104#_A(x1,x2,x3) = x1
    mark_A(x1) = x1 + 1
    active_A(x1) = max{0, x1 - 1}

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U104#(X1,active(X2),X3) -> U104#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U104#(X1,active(X2),X3) -> U104#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U104#_A(x1,x2,x3) = x2
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U105#(mark(X1),X2) -> U105#(X1,X2)
p2: U105#(X1,active(X2)) -> U105#(X1,X2)
p3: U105#(active(X1),X2) -> U105#(X1,X2)
p4: U105#(X1,mark(X2)) -> U105#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U105#_A(x1,x2) = x2
    mark_A(x1) = x1 + 1
    active_A(x1) = x1

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U105#(mark(X1),X2) -> U105#(X1,X2)
p2: U105#(X1,active(X2)) -> U105#(X1,X2)
p3: U105#(active(X1),X2) -> U105#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U105#(mark(X1),X2) -> U105#(X1,X2)
p2: U105#(active(X1),X2) -> U105#(X1,X2)
p3: U105#(X1,active(X2)) -> U105#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U105#_A(x1,x2) = x2 + 1
    mark_A(x1) = max{0, x1 - 1}
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p3

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U105#(mark(X1),X2) -> U105#(X1,X2)
p2: U105#(active(X1),X2) -> U105#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U105#(mark(X1),X2) -> U105#(X1,X2)
p2: U105#(active(X1),X2) -> U105#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U105#_A(x1,x2) = x1
    mark_A(x1) = x1
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U105#(mark(X1),X2) -> U105#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U105#(mark(X1),X2) -> U105#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U105#_A(x1,x2) = x1
    mark_A(x1) = x1 + 1

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: isNat#(mark(X)) -> isNat#(X)
p2: isNat#(active(X)) -> isNat#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    isNat#_A(x1) = max{0, x1 - 1}
    mark_A(x1) = x1 + 1
    active_A(x1) = x1 + 2

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: isNat#(mark(X)) -> isNat#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: isNat#(mark(X)) -> isNat#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  max/plus interpretations on natural numbers:
  
    isNat#_A(x1) = x1
    mark_A(x1) = x1 + 1

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U106#(mark(X)) -> U106#(X)
p2: U106#(active(X)) -> U106#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U106#_A(x1) = max{0, x1 - 1}
    mark_A(x1) = x1 + 1
    active_A(x1) = x1 + 2

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U106#(mark(X)) -> U106#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U106#(mark(X)) -> U106#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  max/plus interpretations on natural numbers:
  
    U106#_A(x1) = x1
    mark_A(x1) = x1 + 1

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: isNatIList#(mark(X)) -> isNatIList#(X)
p2: isNatIList#(active(X)) -> isNatIList#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    isNatIList#_A(x1) = max{0, x1 - 1}
    mark_A(x1) = x1 + 1
    active_A(x1) = x1 + 2

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: isNatIList#(mark(X)) -> isNatIList#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: isNatIList#(mark(X)) -> isNatIList#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  max/plus interpretations on natural numbers:
  
    isNatIList#_A(x1) = x1
    mark_A(x1) = x1 + 1

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U12#(mark(X1),X2) -> U12#(X1,X2)
p2: U12#(X1,active(X2)) -> U12#(X1,X2)
p3: U12#(active(X1),X2) -> U12#(X1,X2)
p4: U12#(X1,mark(X2)) -> U12#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U12#_A(x1,x2) = x2
    mark_A(x1) = x1 + 1
    active_A(x1) = x1

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U12#(mark(X1),X2) -> U12#(X1,X2)
p2: U12#(X1,active(X2)) -> U12#(X1,X2)
p3: U12#(active(X1),X2) -> U12#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U12#(mark(X1),X2) -> U12#(X1,X2)
p2: U12#(active(X1),X2) -> U12#(X1,X2)
p3: U12#(X1,active(X2)) -> U12#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U12#_A(x1,x2) = x2 + 1
    mark_A(x1) = max{0, x1 - 1}
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p3

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U12#(mark(X1),X2) -> U12#(X1,X2)
p2: U12#(active(X1),X2) -> U12#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U12#(mark(X1),X2) -> U12#(X1,X2)
p2: U12#(active(X1),X2) -> U12#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U12#_A(x1,x2) = x1
    mark_A(x1) = x1
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U12#(mark(X1),X2) -> U12#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U12#(mark(X1),X2) -> U12#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U12#_A(x1,x2) = x1
    mark_A(x1) = x1 + 1

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U112#(mark(X1),X2,X3) -> U112#(X1,X2,X3)
p2: U112#(X1,X2,active(X3)) -> U112#(X1,X2,X3)
p3: U112#(X1,active(X2),X3) -> U112#(X1,X2,X3)
p4: U112#(active(X1),X2,X3) -> U112#(X1,X2,X3)
p5: U112#(X1,X2,mark(X3)) -> U112#(X1,X2,X3)
p6: U112#(X1,mark(X2),X3) -> U112#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U112#_A(x1,x2,x3) = x3 + 1
    mark_A(x1) = x1 + 1
    active_A(x1) = x1

The next rules are strictly ordered:

  p5

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U112#(mark(X1),X2,X3) -> U112#(X1,X2,X3)
p2: U112#(X1,X2,active(X3)) -> U112#(X1,X2,X3)
p3: U112#(X1,active(X2),X3) -> U112#(X1,X2,X3)
p4: U112#(active(X1),X2,X3) -> U112#(X1,X2,X3)
p5: U112#(X1,mark(X2),X3) -> U112#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U112#(mark(X1),X2,X3) -> U112#(X1,X2,X3)
p2: U112#(X1,mark(X2),X3) -> U112#(X1,X2,X3)
p3: U112#(active(X1),X2,X3) -> U112#(X1,X2,X3)
p4: U112#(X1,active(X2),X3) -> U112#(X1,X2,X3)
p5: U112#(X1,X2,active(X3)) -> U112#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U112#_A(x1,x2,x3) = x3
    mark_A(x1) = x1 + 2
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p5

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U112#(mark(X1),X2,X3) -> U112#(X1,X2,X3)
p2: U112#(X1,mark(X2),X3) -> U112#(X1,X2,X3)
p3: U112#(active(X1),X2,X3) -> U112#(X1,X2,X3)
p4: U112#(X1,active(X2),X3) -> U112#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U112#(mark(X1),X2,X3) -> U112#(X1,X2,X3)
p2: U112#(X1,active(X2),X3) -> U112#(X1,X2,X3)
p3: U112#(active(X1),X2,X3) -> U112#(X1,X2,X3)
p4: U112#(X1,mark(X2),X3) -> U112#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U112#_A(x1,x2,x3) = max{0, x2 - 1}
    mark_A(x1) = x1 + 2
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U112#(mark(X1),X2,X3) -> U112#(X1,X2,X3)
p2: U112#(X1,active(X2),X3) -> U112#(X1,X2,X3)
p3: U112#(active(X1),X2,X3) -> U112#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U112#(mark(X1),X2,X3) -> U112#(X1,X2,X3)
p2: U112#(active(X1),X2,X3) -> U112#(X1,X2,X3)
p3: U112#(X1,active(X2),X3) -> U112#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U112#_A(x1,x2,x3) = x1 + 1
    mark_A(x1) = x1
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U112#(mark(X1),X2,X3) -> U112#(X1,X2,X3)
p2: U112#(X1,active(X2),X3) -> U112#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U112#(mark(X1),X2,X3) -> U112#(X1,X2,X3)
p2: U112#(X1,active(X2),X3) -> U112#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U112#_A(x1,x2,x3) = x1
    mark_A(x1) = x1 + 1
    active_A(x1) = max{0, x1 - 1}

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U112#(X1,active(X2),X3) -> U112#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U112#(X1,active(X2),X3) -> U112#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U112#_A(x1,x2,x3) = x2
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U113#(mark(X1),X2,X3) -> U113#(X1,X2,X3)
p2: U113#(X1,X2,active(X3)) -> U113#(X1,X2,X3)
p3: U113#(X1,active(X2),X3) -> U113#(X1,X2,X3)
p4: U113#(active(X1),X2,X3) -> U113#(X1,X2,X3)
p5: U113#(X1,X2,mark(X3)) -> U113#(X1,X2,X3)
p6: U113#(X1,mark(X2),X3) -> U113#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U113#_A(x1,x2,x3) = x3 + 1
    mark_A(x1) = x1 + 1
    active_A(x1) = x1

The next rules are strictly ordered:

  p5

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U113#(mark(X1),X2,X3) -> U113#(X1,X2,X3)
p2: U113#(X1,X2,active(X3)) -> U113#(X1,X2,X3)
p3: U113#(X1,active(X2),X3) -> U113#(X1,X2,X3)
p4: U113#(active(X1),X2,X3) -> U113#(X1,X2,X3)
p5: U113#(X1,mark(X2),X3) -> U113#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U113#(mark(X1),X2,X3) -> U113#(X1,X2,X3)
p2: U113#(X1,mark(X2),X3) -> U113#(X1,X2,X3)
p3: U113#(active(X1),X2,X3) -> U113#(X1,X2,X3)
p4: U113#(X1,active(X2),X3) -> U113#(X1,X2,X3)
p5: U113#(X1,X2,active(X3)) -> U113#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U113#_A(x1,x2,x3) = x3
    mark_A(x1) = x1 + 2
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p5

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U113#(mark(X1),X2,X3) -> U113#(X1,X2,X3)
p2: U113#(X1,mark(X2),X3) -> U113#(X1,X2,X3)
p3: U113#(active(X1),X2,X3) -> U113#(X1,X2,X3)
p4: U113#(X1,active(X2),X3) -> U113#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U113#(mark(X1),X2,X3) -> U113#(X1,X2,X3)
p2: U113#(X1,active(X2),X3) -> U113#(X1,X2,X3)
p3: U113#(active(X1),X2,X3) -> U113#(X1,X2,X3)
p4: U113#(X1,mark(X2),X3) -> U113#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U113#_A(x1,x2,x3) = max{2, x2 + 1}
    mark_A(x1) = x1 + 2
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U113#(mark(X1),X2,X3) -> U113#(X1,X2,X3)
p2: U113#(X1,active(X2),X3) -> U113#(X1,X2,X3)
p3: U113#(active(X1),X2,X3) -> U113#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U113#(mark(X1),X2,X3) -> U113#(X1,X2,X3)
p2: U113#(active(X1),X2,X3) -> U113#(X1,X2,X3)
p3: U113#(X1,active(X2),X3) -> U113#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U113#_A(x1,x2,x3) = x1 + 1
    mark_A(x1) = x1
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U113#(mark(X1),X2,X3) -> U113#(X1,X2,X3)
p2: U113#(X1,active(X2),X3) -> U113#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U113#(mark(X1),X2,X3) -> U113#(X1,X2,X3)
p2: U113#(X1,active(X2),X3) -> U113#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U113#_A(x1,x2,x3) = x1
    mark_A(x1) = x1 + 1
    active_A(x1) = max{0, x1 - 1}

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U113#(X1,active(X2),X3) -> U113#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U113#(X1,active(X2),X3) -> U113#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U113#_A(x1,x2,x3) = x2
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U114#(mark(X1),X2) -> U114#(X1,X2)
p2: U114#(X1,active(X2)) -> U114#(X1,X2)
p3: U114#(active(X1),X2) -> U114#(X1,X2)
p4: U114#(X1,mark(X2)) -> U114#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U114#_A(x1,x2) = x2
    mark_A(x1) = x1 + 1
    active_A(x1) = x1

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U114#(mark(X1),X2) -> U114#(X1,X2)
p2: U114#(X1,active(X2)) -> U114#(X1,X2)
p3: U114#(active(X1),X2) -> U114#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U114#(mark(X1),X2) -> U114#(X1,X2)
p2: U114#(active(X1),X2) -> U114#(X1,X2)
p3: U114#(X1,active(X2)) -> U114#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U114#_A(x1,x2) = x2 + 1
    mark_A(x1) = max{0, x1 - 1}
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p3

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U114#(mark(X1),X2) -> U114#(X1,X2)
p2: U114#(active(X1),X2) -> U114#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U114#(mark(X1),X2) -> U114#(X1,X2)
p2: U114#(active(X1),X2) -> U114#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U114#_A(x1,x2) = x1
    mark_A(x1) = x1
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U114#(mark(X1),X2) -> U114#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U114#(mark(X1),X2) -> U114#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U114#_A(x1,x2) = x1
    mark_A(x1) = x1 + 1

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: s#(mark(X)) -> s#(X)
p2: s#(active(X)) -> s#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    s#_A(x1) = max{0, x1 - 1}
    mark_A(x1) = x1 + 1
    active_A(x1) = x1 + 2

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: s#(mark(X)) -> s#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: s#(mark(X)) -> s#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  max/plus interpretations on natural numbers:
  
    s#_A(x1) = x1
    mark_A(x1) = x1 + 1

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: length#(mark(X)) -> length#(X)
p2: length#(active(X)) -> length#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    length#_A(x1) = max{0, x1 - 1}
    mark_A(x1) = x1 + 1
    active_A(x1) = x1 + 2

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: length#(mark(X)) -> length#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: length#(mark(X)) -> length#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  max/plus interpretations on natural numbers:
  
    length#_A(x1) = x1
    mark_A(x1) = x1 + 1

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U13#(mark(X)) -> U13#(X)
p2: U13#(active(X)) -> U13#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U13#_A(x1) = max{0, x1 - 1}
    mark_A(x1) = x1 + 1
    active_A(x1) = x1 + 2

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U13#(mark(X)) -> U13#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U13#(mark(X)) -> U13#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  max/plus interpretations on natural numbers:
  
    U13#_A(x1) = x1
    mark_A(x1) = x1 + 1

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: isNatList#(mark(X)) -> isNatList#(X)
p2: isNatList#(active(X)) -> isNatList#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    isNatList#_A(x1) = max{0, x1 - 1}
    mark_A(x1) = x1 + 1
    active_A(x1) = x1 + 2

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: isNatList#(mark(X)) -> isNatList#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: isNatList#(mark(X)) -> isNatList#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  max/plus interpretations on natural numbers:
  
    isNatList#_A(x1) = x1
    mark_A(x1) = x1 + 1

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U122#(mark(X)) -> U122#(X)
p2: U122#(active(X)) -> U122#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U122#_A(x1) = max{0, x1 - 1}
    mark_A(x1) = x1 + 1
    active_A(x1) = x1 + 2

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U122#(mark(X)) -> U122#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U122#(mark(X)) -> U122#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  max/plus interpretations on natural numbers:
  
    U122#_A(x1) = x1
    mark_A(x1) = x1 + 1

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U132#(mark(X1),X2,X3,X4) -> U132#(X1,X2,X3,X4)
p2: U132#(X1,X2,X3,active(X4)) -> U132#(X1,X2,X3,X4)
p3: U132#(X1,X2,active(X3),X4) -> U132#(X1,X2,X3,X4)
p4: U132#(X1,active(X2),X3,X4) -> U132#(X1,X2,X3,X4)
p5: U132#(active(X1),X2,X3,X4) -> U132#(X1,X2,X3,X4)
p6: U132#(X1,X2,X3,mark(X4)) -> U132#(X1,X2,X3,X4)
p7: U132#(X1,X2,mark(X3),X4) -> U132#(X1,X2,X3,X4)
p8: U132#(X1,mark(X2),X3,X4) -> U132#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U132#_A(x1,x2,x3,x4) = max{0, x4 - 1}
    mark_A(x1) = x1 + 2
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p6

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U132#(mark(X1),X2,X3,X4) -> U132#(X1,X2,X3,X4)
p2: U132#(X1,X2,X3,active(X4)) -> U132#(X1,X2,X3,X4)
p3: U132#(X1,X2,active(X3),X4) -> U132#(X1,X2,X3,X4)
p4: U132#(X1,active(X2),X3,X4) -> U132#(X1,X2,X3,X4)
p5: U132#(active(X1),X2,X3,X4) -> U132#(X1,X2,X3,X4)
p6: U132#(X1,X2,mark(X3),X4) -> U132#(X1,X2,X3,X4)
p7: U132#(X1,mark(X2),X3,X4) -> U132#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U132#(mark(X1),X2,X3,X4) -> U132#(X1,X2,X3,X4)
p2: U132#(X1,mark(X2),X3,X4) -> U132#(X1,X2,X3,X4)
p3: U132#(X1,X2,mark(X3),X4) -> U132#(X1,X2,X3,X4)
p4: U132#(active(X1),X2,X3,X4) -> U132#(X1,X2,X3,X4)
p5: U132#(X1,active(X2),X3,X4) -> U132#(X1,X2,X3,X4)
p6: U132#(X1,X2,active(X3),X4) -> U132#(X1,X2,X3,X4)
p7: U132#(X1,X2,X3,active(X4)) -> U132#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U132#_A(x1,x2,x3,x4) = x1 + 1
    mark_A(x1) = x1
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U132#(mark(X1),X2,X3,X4) -> U132#(X1,X2,X3,X4)
p2: U132#(X1,mark(X2),X3,X4) -> U132#(X1,X2,X3,X4)
p3: U132#(X1,X2,mark(X3),X4) -> U132#(X1,X2,X3,X4)
p4: U132#(X1,active(X2),X3,X4) -> U132#(X1,X2,X3,X4)
p5: U132#(X1,X2,active(X3),X4) -> U132#(X1,X2,X3,X4)
p6: U132#(X1,X2,X3,active(X4)) -> U132#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U132#(mark(X1),X2,X3,X4) -> U132#(X1,X2,X3,X4)
p2: U132#(X1,X2,X3,active(X4)) -> U132#(X1,X2,X3,X4)
p3: U132#(X1,X2,active(X3),X4) -> U132#(X1,X2,X3,X4)
p4: U132#(X1,active(X2),X3,X4) -> U132#(X1,X2,X3,X4)
p5: U132#(X1,X2,mark(X3),X4) -> U132#(X1,X2,X3,X4)
p6: U132#(X1,mark(X2),X3,X4) -> U132#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U132#_A(x1,x2,x3,x4) = x1 + 3
    mark_A(x1) = x1 + 1
    active_A(x1) = max{2, x1 - 1}

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U132#(X1,X2,X3,active(X4)) -> U132#(X1,X2,X3,X4)
p2: U132#(X1,X2,active(X3),X4) -> U132#(X1,X2,X3,X4)
p3: U132#(X1,active(X2),X3,X4) -> U132#(X1,X2,X3,X4)
p4: U132#(X1,X2,mark(X3),X4) -> U132#(X1,X2,X3,X4)
p5: U132#(X1,mark(X2),X3,X4) -> U132#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U132#(X1,X2,X3,active(X4)) -> U132#(X1,X2,X3,X4)
p2: U132#(X1,mark(X2),X3,X4) -> U132#(X1,X2,X3,X4)
p3: U132#(X1,X2,mark(X3),X4) -> U132#(X1,X2,X3,X4)
p4: U132#(X1,active(X2),X3,X4) -> U132#(X1,X2,X3,X4)
p5: U132#(X1,X2,active(X3),X4) -> U132#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U132#_A(x1,x2,x3,x4) = x4
    active_A(x1) = x1 + 1
    mark_A(x1) = max{0, x1 - 1}

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U132#(X1,mark(X2),X3,X4) -> U132#(X1,X2,X3,X4)
p2: U132#(X1,X2,mark(X3),X4) -> U132#(X1,X2,X3,X4)
p3: U132#(X1,active(X2),X3,X4) -> U132#(X1,X2,X3,X4)
p4: U132#(X1,X2,active(X3),X4) -> U132#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U132#(X1,mark(X2),X3,X4) -> U132#(X1,X2,X3,X4)
p2: U132#(X1,X2,active(X3),X4) -> U132#(X1,X2,X3,X4)
p3: U132#(X1,active(X2),X3,X4) -> U132#(X1,X2,X3,X4)
p4: U132#(X1,X2,mark(X3),X4) -> U132#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U132#_A(x1,x2,x3,x4) = max{0, x3 - 1}
    mark_A(x1) = x1 + 2
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U132#(X1,mark(X2),X3,X4) -> U132#(X1,X2,X3,X4)
p2: U132#(X1,X2,active(X3),X4) -> U132#(X1,X2,X3,X4)
p3: U132#(X1,active(X2),X3,X4) -> U132#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U132#(X1,mark(X2),X3,X4) -> U132#(X1,X2,X3,X4)
p2: U132#(X1,active(X2),X3,X4) -> U132#(X1,X2,X3,X4)
p3: U132#(X1,X2,active(X3),X4) -> U132#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U132#_A(x1,x2,x3,x4) = x2 + 1
    mark_A(x1) = x1
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U132#(X1,mark(X2),X3,X4) -> U132#(X1,X2,X3,X4)
p2: U132#(X1,X2,active(X3),X4) -> U132#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U132#(X1,mark(X2),X3,X4) -> U132#(X1,X2,X3,X4)
p2: U132#(X1,X2,active(X3),X4) -> U132#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U132#_A(x1,x2,x3,x4) = x3
    mark_A(x1) = max{0, x1 - 1}
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U132#(X1,mark(X2),X3,X4) -> U132#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U132#(X1,mark(X2),X3,X4) -> U132#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U132#_A(x1,x2,x3,x4) = x2
    mark_A(x1) = x1 + 1

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U133#(mark(X1),X2,X3,X4) -> U133#(X1,X2,X3,X4)
p2: U133#(X1,X2,X3,active(X4)) -> U133#(X1,X2,X3,X4)
p3: U133#(X1,X2,active(X3),X4) -> U133#(X1,X2,X3,X4)
p4: U133#(X1,active(X2),X3,X4) -> U133#(X1,X2,X3,X4)
p5: U133#(active(X1),X2,X3,X4) -> U133#(X1,X2,X3,X4)
p6: U133#(X1,X2,X3,mark(X4)) -> U133#(X1,X2,X3,X4)
p7: U133#(X1,X2,mark(X3),X4) -> U133#(X1,X2,X3,X4)
p8: U133#(X1,mark(X2),X3,X4) -> U133#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U133#_A(x1,x2,x3,x4) = max{0, x4 - 1}
    mark_A(x1) = x1 + 2
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p6

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U133#(mark(X1),X2,X3,X4) -> U133#(X1,X2,X3,X4)
p2: U133#(X1,X2,X3,active(X4)) -> U133#(X1,X2,X3,X4)
p3: U133#(X1,X2,active(X3),X4) -> U133#(X1,X2,X3,X4)
p4: U133#(X1,active(X2),X3,X4) -> U133#(X1,X2,X3,X4)
p5: U133#(active(X1),X2,X3,X4) -> U133#(X1,X2,X3,X4)
p6: U133#(X1,X2,mark(X3),X4) -> U133#(X1,X2,X3,X4)
p7: U133#(X1,mark(X2),X3,X4) -> U133#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U133#(mark(X1),X2,X3,X4) -> U133#(X1,X2,X3,X4)
p2: U133#(X1,mark(X2),X3,X4) -> U133#(X1,X2,X3,X4)
p3: U133#(X1,X2,mark(X3),X4) -> U133#(X1,X2,X3,X4)
p4: U133#(active(X1),X2,X3,X4) -> U133#(X1,X2,X3,X4)
p5: U133#(X1,active(X2),X3,X4) -> U133#(X1,X2,X3,X4)
p6: U133#(X1,X2,active(X3),X4) -> U133#(X1,X2,X3,X4)
p7: U133#(X1,X2,X3,active(X4)) -> U133#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U133#_A(x1,x2,x3,x4) = x1 + 1
    mark_A(x1) = x1
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U133#(mark(X1),X2,X3,X4) -> U133#(X1,X2,X3,X4)
p2: U133#(X1,mark(X2),X3,X4) -> U133#(X1,X2,X3,X4)
p3: U133#(X1,X2,mark(X3),X4) -> U133#(X1,X2,X3,X4)
p4: U133#(X1,active(X2),X3,X4) -> U133#(X1,X2,X3,X4)
p5: U133#(X1,X2,active(X3),X4) -> U133#(X1,X2,X3,X4)
p6: U133#(X1,X2,X3,active(X4)) -> U133#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U133#(mark(X1),X2,X3,X4) -> U133#(X1,X2,X3,X4)
p2: U133#(X1,X2,X3,active(X4)) -> U133#(X1,X2,X3,X4)
p3: U133#(X1,X2,active(X3),X4) -> U133#(X1,X2,X3,X4)
p4: U133#(X1,active(X2),X3,X4) -> U133#(X1,X2,X3,X4)
p5: U133#(X1,X2,mark(X3),X4) -> U133#(X1,X2,X3,X4)
p6: U133#(X1,mark(X2),X3,X4) -> U133#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U133#_A(x1,x2,x3,x4) = x1 + 3
    mark_A(x1) = x1 + 1
    active_A(x1) = max{2, x1 - 1}

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U133#(X1,X2,X3,active(X4)) -> U133#(X1,X2,X3,X4)
p2: U133#(X1,X2,active(X3),X4) -> U133#(X1,X2,X3,X4)
p3: U133#(X1,active(X2),X3,X4) -> U133#(X1,X2,X3,X4)
p4: U133#(X1,X2,mark(X3),X4) -> U133#(X1,X2,X3,X4)
p5: U133#(X1,mark(X2),X3,X4) -> U133#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U133#(X1,X2,X3,active(X4)) -> U133#(X1,X2,X3,X4)
p2: U133#(X1,mark(X2),X3,X4) -> U133#(X1,X2,X3,X4)
p3: U133#(X1,X2,mark(X3),X4) -> U133#(X1,X2,X3,X4)
p4: U133#(X1,active(X2),X3,X4) -> U133#(X1,X2,X3,X4)
p5: U133#(X1,X2,active(X3),X4) -> U133#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U133#_A(x1,x2,x3,x4) = x4
    active_A(x1) = x1 + 1
    mark_A(x1) = max{0, x1 - 1}

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U133#(X1,mark(X2),X3,X4) -> U133#(X1,X2,X3,X4)
p2: U133#(X1,X2,mark(X3),X4) -> U133#(X1,X2,X3,X4)
p3: U133#(X1,active(X2),X3,X4) -> U133#(X1,X2,X3,X4)
p4: U133#(X1,X2,active(X3),X4) -> U133#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U133#(X1,mark(X2),X3,X4) -> U133#(X1,X2,X3,X4)
p2: U133#(X1,X2,active(X3),X4) -> U133#(X1,X2,X3,X4)
p3: U133#(X1,active(X2),X3,X4) -> U133#(X1,X2,X3,X4)
p4: U133#(X1,X2,mark(X3),X4) -> U133#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U133#_A(x1,x2,x3,x4) = max{0, x3 - 1}
    mark_A(x1) = x1 + 2
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U133#(X1,mark(X2),X3,X4) -> U133#(X1,X2,X3,X4)
p2: U133#(X1,X2,active(X3),X4) -> U133#(X1,X2,X3,X4)
p3: U133#(X1,active(X2),X3,X4) -> U133#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U133#(X1,mark(X2),X3,X4) -> U133#(X1,X2,X3,X4)
p2: U133#(X1,active(X2),X3,X4) -> U133#(X1,X2,X3,X4)
p3: U133#(X1,X2,active(X3),X4) -> U133#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U133#_A(x1,x2,x3,x4) = x2 + 1
    mark_A(x1) = x1
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U133#(X1,mark(X2),X3,X4) -> U133#(X1,X2,X3,X4)
p2: U133#(X1,X2,active(X3),X4) -> U133#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U133#(X1,mark(X2),X3,X4) -> U133#(X1,X2,X3,X4)
p2: U133#(X1,X2,active(X3),X4) -> U133#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U133#_A(x1,x2,x3,x4) = x3
    mark_A(x1) = max{0, x1 - 1}
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U133#(X1,mark(X2),X3,X4) -> U133#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U133#(X1,mark(X2),X3,X4) -> U133#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U133#_A(x1,x2,x3,x4) = x2
    mark_A(x1) = x1 + 1

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U134#(mark(X1),X2,X3,X4) -> U134#(X1,X2,X3,X4)
p2: U134#(X1,X2,X3,active(X4)) -> U134#(X1,X2,X3,X4)
p3: U134#(X1,X2,active(X3),X4) -> U134#(X1,X2,X3,X4)
p4: U134#(X1,active(X2),X3,X4) -> U134#(X1,X2,X3,X4)
p5: U134#(active(X1),X2,X3,X4) -> U134#(X1,X2,X3,X4)
p6: U134#(X1,X2,X3,mark(X4)) -> U134#(X1,X2,X3,X4)
p7: U134#(X1,X2,mark(X3),X4) -> U134#(X1,X2,X3,X4)
p8: U134#(X1,mark(X2),X3,X4) -> U134#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U134#_A(x1,x2,x3,x4) = max{0, x4 - 1}
    mark_A(x1) = x1 + 2
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p6

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U134#(mark(X1),X2,X3,X4) -> U134#(X1,X2,X3,X4)
p2: U134#(X1,X2,X3,active(X4)) -> U134#(X1,X2,X3,X4)
p3: U134#(X1,X2,active(X3),X4) -> U134#(X1,X2,X3,X4)
p4: U134#(X1,active(X2),X3,X4) -> U134#(X1,X2,X3,X4)
p5: U134#(active(X1),X2,X3,X4) -> U134#(X1,X2,X3,X4)
p6: U134#(X1,X2,mark(X3),X4) -> U134#(X1,X2,X3,X4)
p7: U134#(X1,mark(X2),X3,X4) -> U134#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U134#(mark(X1),X2,X3,X4) -> U134#(X1,X2,X3,X4)
p2: U134#(X1,mark(X2),X3,X4) -> U134#(X1,X2,X3,X4)
p3: U134#(X1,X2,mark(X3),X4) -> U134#(X1,X2,X3,X4)
p4: U134#(active(X1),X2,X3,X4) -> U134#(X1,X2,X3,X4)
p5: U134#(X1,active(X2),X3,X4) -> U134#(X1,X2,X3,X4)
p6: U134#(X1,X2,active(X3),X4) -> U134#(X1,X2,X3,X4)
p7: U134#(X1,X2,X3,active(X4)) -> U134#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U134#_A(x1,x2,x3,x4) = x1 + 1
    mark_A(x1) = x1
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U134#(mark(X1),X2,X3,X4) -> U134#(X1,X2,X3,X4)
p2: U134#(X1,mark(X2),X3,X4) -> U134#(X1,X2,X3,X4)
p3: U134#(X1,X2,mark(X3),X4) -> U134#(X1,X2,X3,X4)
p4: U134#(X1,active(X2),X3,X4) -> U134#(X1,X2,X3,X4)
p5: U134#(X1,X2,active(X3),X4) -> U134#(X1,X2,X3,X4)
p6: U134#(X1,X2,X3,active(X4)) -> U134#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U134#(mark(X1),X2,X3,X4) -> U134#(X1,X2,X3,X4)
p2: U134#(X1,X2,X3,active(X4)) -> U134#(X1,X2,X3,X4)
p3: U134#(X1,X2,active(X3),X4) -> U134#(X1,X2,X3,X4)
p4: U134#(X1,active(X2),X3,X4) -> U134#(X1,X2,X3,X4)
p5: U134#(X1,X2,mark(X3),X4) -> U134#(X1,X2,X3,X4)
p6: U134#(X1,mark(X2),X3,X4) -> U134#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U134#_A(x1,x2,x3,x4) = x1 + 3
    mark_A(x1) = x1 + 1
    active_A(x1) = max{2, x1 - 1}

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U134#(X1,X2,X3,active(X4)) -> U134#(X1,X2,X3,X4)
p2: U134#(X1,X2,active(X3),X4) -> U134#(X1,X2,X3,X4)
p3: U134#(X1,active(X2),X3,X4) -> U134#(X1,X2,X3,X4)
p4: U134#(X1,X2,mark(X3),X4) -> U134#(X1,X2,X3,X4)
p5: U134#(X1,mark(X2),X3,X4) -> U134#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U134#(X1,X2,X3,active(X4)) -> U134#(X1,X2,X3,X4)
p2: U134#(X1,mark(X2),X3,X4) -> U134#(X1,X2,X3,X4)
p3: U134#(X1,X2,mark(X3),X4) -> U134#(X1,X2,X3,X4)
p4: U134#(X1,active(X2),X3,X4) -> U134#(X1,X2,X3,X4)
p5: U134#(X1,X2,active(X3),X4) -> U134#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U134#_A(x1,x2,x3,x4) = x4
    active_A(x1) = x1 + 1
    mark_A(x1) = max{0, x1 - 1}

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U134#(X1,mark(X2),X3,X4) -> U134#(X1,X2,X3,X4)
p2: U134#(X1,X2,mark(X3),X4) -> U134#(X1,X2,X3,X4)
p3: U134#(X1,active(X2),X3,X4) -> U134#(X1,X2,X3,X4)
p4: U134#(X1,X2,active(X3),X4) -> U134#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U134#(X1,mark(X2),X3,X4) -> U134#(X1,X2,X3,X4)
p2: U134#(X1,X2,active(X3),X4) -> U134#(X1,X2,X3,X4)
p3: U134#(X1,active(X2),X3,X4) -> U134#(X1,X2,X3,X4)
p4: U134#(X1,X2,mark(X3),X4) -> U134#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U134#_A(x1,x2,x3,x4) = max{0, x3 - 1}
    mark_A(x1) = x1 + 2
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U134#(X1,mark(X2),X3,X4) -> U134#(X1,X2,X3,X4)
p2: U134#(X1,X2,active(X3),X4) -> U134#(X1,X2,X3,X4)
p3: U134#(X1,active(X2),X3,X4) -> U134#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U134#(X1,mark(X2),X3,X4) -> U134#(X1,X2,X3,X4)
p2: U134#(X1,active(X2),X3,X4) -> U134#(X1,X2,X3,X4)
p3: U134#(X1,X2,active(X3),X4) -> U134#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U134#_A(x1,x2,x3,x4) = x2 + 1
    mark_A(x1) = x1
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U134#(X1,mark(X2),X3,X4) -> U134#(X1,X2,X3,X4)
p2: U134#(X1,X2,active(X3),X4) -> U134#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U134#(X1,mark(X2),X3,X4) -> U134#(X1,X2,X3,X4)
p2: U134#(X1,X2,active(X3),X4) -> U134#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U134#_A(x1,x2,x3,x4) = x3
    mark_A(x1) = max{0, x1 - 1}
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U134#(X1,mark(X2),X3,X4) -> U134#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U134#(X1,mark(X2),X3,X4) -> U134#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U134#_A(x1,x2,x3,x4) = x2
    mark_A(x1) = x1 + 1

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U135#(mark(X1),X2,X3,X4) -> U135#(X1,X2,X3,X4)
p2: U135#(X1,X2,X3,active(X4)) -> U135#(X1,X2,X3,X4)
p3: U135#(X1,X2,active(X3),X4) -> U135#(X1,X2,X3,X4)
p4: U135#(X1,active(X2),X3,X4) -> U135#(X1,X2,X3,X4)
p5: U135#(active(X1),X2,X3,X4) -> U135#(X1,X2,X3,X4)
p6: U135#(X1,X2,X3,mark(X4)) -> U135#(X1,X2,X3,X4)
p7: U135#(X1,X2,mark(X3),X4) -> U135#(X1,X2,X3,X4)
p8: U135#(X1,mark(X2),X3,X4) -> U135#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U135#_A(x1,x2,x3,x4) = max{0, x4 - 1}
    mark_A(x1) = x1 + 2
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p6

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U135#(mark(X1),X2,X3,X4) -> U135#(X1,X2,X3,X4)
p2: U135#(X1,X2,X3,active(X4)) -> U135#(X1,X2,X3,X4)
p3: U135#(X1,X2,active(X3),X4) -> U135#(X1,X2,X3,X4)
p4: U135#(X1,active(X2),X3,X4) -> U135#(X1,X2,X3,X4)
p5: U135#(active(X1),X2,X3,X4) -> U135#(X1,X2,X3,X4)
p6: U135#(X1,X2,mark(X3),X4) -> U135#(X1,X2,X3,X4)
p7: U135#(X1,mark(X2),X3,X4) -> U135#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U135#(mark(X1),X2,X3,X4) -> U135#(X1,X2,X3,X4)
p2: U135#(X1,mark(X2),X3,X4) -> U135#(X1,X2,X3,X4)
p3: U135#(X1,X2,mark(X3),X4) -> U135#(X1,X2,X3,X4)
p4: U135#(active(X1),X2,X3,X4) -> U135#(X1,X2,X3,X4)
p5: U135#(X1,active(X2),X3,X4) -> U135#(X1,X2,X3,X4)
p6: U135#(X1,X2,active(X3),X4) -> U135#(X1,X2,X3,X4)
p7: U135#(X1,X2,X3,active(X4)) -> U135#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U135#_A(x1,x2,x3,x4) = x1 + 1
    mark_A(x1) = x1
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U135#(mark(X1),X2,X3,X4) -> U135#(X1,X2,X3,X4)
p2: U135#(X1,mark(X2),X3,X4) -> U135#(X1,X2,X3,X4)
p3: U135#(X1,X2,mark(X3),X4) -> U135#(X1,X2,X3,X4)
p4: U135#(X1,active(X2),X3,X4) -> U135#(X1,X2,X3,X4)
p5: U135#(X1,X2,active(X3),X4) -> U135#(X1,X2,X3,X4)
p6: U135#(X1,X2,X3,active(X4)) -> U135#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U135#(mark(X1),X2,X3,X4) -> U135#(X1,X2,X3,X4)
p2: U135#(X1,X2,X3,active(X4)) -> U135#(X1,X2,X3,X4)
p3: U135#(X1,X2,active(X3),X4) -> U135#(X1,X2,X3,X4)
p4: U135#(X1,active(X2),X3,X4) -> U135#(X1,X2,X3,X4)
p5: U135#(X1,X2,mark(X3),X4) -> U135#(X1,X2,X3,X4)
p6: U135#(X1,mark(X2),X3,X4) -> U135#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U135#_A(x1,x2,x3,x4) = x1 + 3
    mark_A(x1) = x1 + 1
    active_A(x1) = max{2, x1 - 1}

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U135#(X1,X2,X3,active(X4)) -> U135#(X1,X2,X3,X4)
p2: U135#(X1,X2,active(X3),X4) -> U135#(X1,X2,X3,X4)
p3: U135#(X1,active(X2),X3,X4) -> U135#(X1,X2,X3,X4)
p4: U135#(X1,X2,mark(X3),X4) -> U135#(X1,X2,X3,X4)
p5: U135#(X1,mark(X2),X3,X4) -> U135#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U135#(X1,X2,X3,active(X4)) -> U135#(X1,X2,X3,X4)
p2: U135#(X1,mark(X2),X3,X4) -> U135#(X1,X2,X3,X4)
p3: U135#(X1,X2,mark(X3),X4) -> U135#(X1,X2,X3,X4)
p4: U135#(X1,active(X2),X3,X4) -> U135#(X1,X2,X3,X4)
p5: U135#(X1,X2,active(X3),X4) -> U135#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U135#_A(x1,x2,x3,x4) = x4
    active_A(x1) = x1 + 1
    mark_A(x1) = max{0, x1 - 1}

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U135#(X1,mark(X2),X3,X4) -> U135#(X1,X2,X3,X4)
p2: U135#(X1,X2,mark(X3),X4) -> U135#(X1,X2,X3,X4)
p3: U135#(X1,active(X2),X3,X4) -> U135#(X1,X2,X3,X4)
p4: U135#(X1,X2,active(X3),X4) -> U135#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U135#(X1,mark(X2),X3,X4) -> U135#(X1,X2,X3,X4)
p2: U135#(X1,X2,active(X3),X4) -> U135#(X1,X2,X3,X4)
p3: U135#(X1,active(X2),X3,X4) -> U135#(X1,X2,X3,X4)
p4: U135#(X1,X2,mark(X3),X4) -> U135#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U135#_A(x1,x2,x3,x4) = max{0, x3 - 1}
    mark_A(x1) = x1 + 2
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U135#(X1,mark(X2),X3,X4) -> U135#(X1,X2,X3,X4)
p2: U135#(X1,X2,active(X3),X4) -> U135#(X1,X2,X3,X4)
p3: U135#(X1,active(X2),X3,X4) -> U135#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U135#(X1,mark(X2),X3,X4) -> U135#(X1,X2,X3,X4)
p2: U135#(X1,active(X2),X3,X4) -> U135#(X1,X2,X3,X4)
p3: U135#(X1,X2,active(X3),X4) -> U135#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U135#_A(x1,x2,x3,x4) = x2 + 1
    mark_A(x1) = x1
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U135#(X1,mark(X2),X3,X4) -> U135#(X1,X2,X3,X4)
p2: U135#(X1,X2,active(X3),X4) -> U135#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U135#(X1,mark(X2),X3,X4) -> U135#(X1,X2,X3,X4)
p2: U135#(X1,X2,active(X3),X4) -> U135#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U135#_A(x1,x2,x3,x4) = x3
    mark_A(x1) = max{0, x1 - 1}
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U135#(X1,mark(X2),X3,X4) -> U135#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U135#(X1,mark(X2),X3,X4) -> U135#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U135#_A(x1,x2,x3,x4) = x2
    mark_A(x1) = x1 + 1

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U136#(mark(X1),X2,X3,X4) -> U136#(X1,X2,X3,X4)
p2: U136#(X1,X2,X3,active(X4)) -> U136#(X1,X2,X3,X4)
p3: U136#(X1,X2,active(X3),X4) -> U136#(X1,X2,X3,X4)
p4: U136#(X1,active(X2),X3,X4) -> U136#(X1,X2,X3,X4)
p5: U136#(active(X1),X2,X3,X4) -> U136#(X1,X2,X3,X4)
p6: U136#(X1,X2,X3,mark(X4)) -> U136#(X1,X2,X3,X4)
p7: U136#(X1,X2,mark(X3),X4) -> U136#(X1,X2,X3,X4)
p8: U136#(X1,mark(X2),X3,X4) -> U136#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U136#_A(x1,x2,x3,x4) = max{0, x4 - 1}
    mark_A(x1) = x1 + 2
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p6

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U136#(mark(X1),X2,X3,X4) -> U136#(X1,X2,X3,X4)
p2: U136#(X1,X2,X3,active(X4)) -> U136#(X1,X2,X3,X4)
p3: U136#(X1,X2,active(X3),X4) -> U136#(X1,X2,X3,X4)
p4: U136#(X1,active(X2),X3,X4) -> U136#(X1,X2,X3,X4)
p5: U136#(active(X1),X2,X3,X4) -> U136#(X1,X2,X3,X4)
p6: U136#(X1,X2,mark(X3),X4) -> U136#(X1,X2,X3,X4)
p7: U136#(X1,mark(X2),X3,X4) -> U136#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U136#(mark(X1),X2,X3,X4) -> U136#(X1,X2,X3,X4)
p2: U136#(X1,mark(X2),X3,X4) -> U136#(X1,X2,X3,X4)
p3: U136#(X1,X2,mark(X3),X4) -> U136#(X1,X2,X3,X4)
p4: U136#(active(X1),X2,X3,X4) -> U136#(X1,X2,X3,X4)
p5: U136#(X1,active(X2),X3,X4) -> U136#(X1,X2,X3,X4)
p6: U136#(X1,X2,active(X3),X4) -> U136#(X1,X2,X3,X4)
p7: U136#(X1,X2,X3,active(X4)) -> U136#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U136#_A(x1,x2,x3,x4) = x1 + 1
    mark_A(x1) = x1
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U136#(mark(X1),X2,X3,X4) -> U136#(X1,X2,X3,X4)
p2: U136#(X1,mark(X2),X3,X4) -> U136#(X1,X2,X3,X4)
p3: U136#(X1,X2,mark(X3),X4) -> U136#(X1,X2,X3,X4)
p4: U136#(X1,active(X2),X3,X4) -> U136#(X1,X2,X3,X4)
p5: U136#(X1,X2,active(X3),X4) -> U136#(X1,X2,X3,X4)
p6: U136#(X1,X2,X3,active(X4)) -> U136#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U136#(mark(X1),X2,X3,X4) -> U136#(X1,X2,X3,X4)
p2: U136#(X1,X2,X3,active(X4)) -> U136#(X1,X2,X3,X4)
p3: U136#(X1,X2,active(X3),X4) -> U136#(X1,X2,X3,X4)
p4: U136#(X1,active(X2),X3,X4) -> U136#(X1,X2,X3,X4)
p5: U136#(X1,X2,mark(X3),X4) -> U136#(X1,X2,X3,X4)
p6: U136#(X1,mark(X2),X3,X4) -> U136#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U136#_A(x1,x2,x3,x4) = x1 + 3
    mark_A(x1) = x1 + 1
    active_A(x1) = max{2, x1 - 1}

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U136#(X1,X2,X3,active(X4)) -> U136#(X1,X2,X3,X4)
p2: U136#(X1,X2,active(X3),X4) -> U136#(X1,X2,X3,X4)
p3: U136#(X1,active(X2),X3,X4) -> U136#(X1,X2,X3,X4)
p4: U136#(X1,X2,mark(X3),X4) -> U136#(X1,X2,X3,X4)
p5: U136#(X1,mark(X2),X3,X4) -> U136#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U136#(X1,X2,X3,active(X4)) -> U136#(X1,X2,X3,X4)
p2: U136#(X1,mark(X2),X3,X4) -> U136#(X1,X2,X3,X4)
p3: U136#(X1,X2,mark(X3),X4) -> U136#(X1,X2,X3,X4)
p4: U136#(X1,active(X2),X3,X4) -> U136#(X1,X2,X3,X4)
p5: U136#(X1,X2,active(X3),X4) -> U136#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U136#_A(x1,x2,x3,x4) = x4
    active_A(x1) = x1 + 1
    mark_A(x1) = max{0, x1 - 1}

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U136#(X1,mark(X2),X3,X4) -> U136#(X1,X2,X3,X4)
p2: U136#(X1,X2,mark(X3),X4) -> U136#(X1,X2,X3,X4)
p3: U136#(X1,active(X2),X3,X4) -> U136#(X1,X2,X3,X4)
p4: U136#(X1,X2,active(X3),X4) -> U136#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U136#(X1,mark(X2),X3,X4) -> U136#(X1,X2,X3,X4)
p2: U136#(X1,X2,active(X3),X4) -> U136#(X1,X2,X3,X4)
p3: U136#(X1,active(X2),X3,X4) -> U136#(X1,X2,X3,X4)
p4: U136#(X1,X2,mark(X3),X4) -> U136#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U136#_A(x1,x2,x3,x4) = max{0, x3 - 1}
    mark_A(x1) = x1 + 2
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U136#(X1,mark(X2),X3,X4) -> U136#(X1,X2,X3,X4)
p2: U136#(X1,X2,active(X3),X4) -> U136#(X1,X2,X3,X4)
p3: U136#(X1,active(X2),X3,X4) -> U136#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U136#(X1,mark(X2),X3,X4) -> U136#(X1,X2,X3,X4)
p2: U136#(X1,active(X2),X3,X4) -> U136#(X1,X2,X3,X4)
p3: U136#(X1,X2,active(X3),X4) -> U136#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U136#_A(x1,x2,x3,x4) = x2 + 1
    mark_A(x1) = x1
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U136#(X1,mark(X2),X3,X4) -> U136#(X1,X2,X3,X4)
p2: U136#(X1,X2,active(X3),X4) -> U136#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U136#(X1,mark(X2),X3,X4) -> U136#(X1,X2,X3,X4)
p2: U136#(X1,X2,active(X3),X4) -> U136#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U136#_A(x1,x2,x3,x4) = x3
    mark_A(x1) = max{0, x1 - 1}
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U136#(X1,mark(X2),X3,X4) -> U136#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U136#(X1,mark(X2),X3,X4) -> U136#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U136#_A(x1,x2,x3,x4) = x2
    mark_A(x1) = x1 + 1

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: cons#(mark(X1),X2) -> cons#(X1,X2)
p2: cons#(X1,active(X2)) -> cons#(X1,X2)
p3: cons#(active(X1),X2) -> cons#(X1,X2)
p4: cons#(X1,mark(X2)) -> cons#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    cons#_A(x1,x2) = x2
    mark_A(x1) = x1 + 1
    active_A(x1) = x1

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: cons#(mark(X1),X2) -> cons#(X1,X2)
p2: cons#(X1,active(X2)) -> cons#(X1,X2)
p3: cons#(active(X1),X2) -> cons#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: cons#(mark(X1),X2) -> cons#(X1,X2)
p2: cons#(active(X1),X2) -> cons#(X1,X2)
p3: cons#(X1,active(X2)) -> cons#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    cons#_A(x1,x2) = x2 + 1
    mark_A(x1) = max{0, x1 - 1}
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p3

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: cons#(mark(X1),X2) -> cons#(X1,X2)
p2: cons#(active(X1),X2) -> cons#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: cons#(mark(X1),X2) -> cons#(X1,X2)
p2: cons#(active(X1),X2) -> cons#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    cons#_A(x1,x2) = x1
    mark_A(x1) = x1
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: cons#(mark(X1),X2) -> cons#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: cons#(mark(X1),X2) -> cons#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    cons#_A(x1,x2) = x1
    mark_A(x1) = x1 + 1

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: take#(mark(X1),X2) -> take#(X1,X2)
p2: take#(X1,active(X2)) -> take#(X1,X2)
p3: take#(active(X1),X2) -> take#(X1,X2)
p4: take#(X1,mark(X2)) -> take#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    take#_A(x1,x2) = x2
    mark_A(x1) = x1 + 1
    active_A(x1) = x1

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: take#(mark(X1),X2) -> take#(X1,X2)
p2: take#(X1,active(X2)) -> take#(X1,X2)
p3: take#(active(X1),X2) -> take#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: take#(mark(X1),X2) -> take#(X1,X2)
p2: take#(active(X1),X2) -> take#(X1,X2)
p3: take#(X1,active(X2)) -> take#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    take#_A(x1,x2) = x2 + 1
    mark_A(x1) = max{0, x1 - 1}
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p3

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: take#(mark(X1),X2) -> take#(X1,X2)
p2: take#(active(X1),X2) -> take#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: take#(mark(X1),X2) -> take#(X1,X2)
p2: take#(active(X1),X2) -> take#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    take#_A(x1,x2) = x1
    mark_A(x1) = x1
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: take#(mark(X1),X2) -> take#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: take#(mark(X1),X2) -> take#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    take#_A(x1,x2) = x1
    mark_A(x1) = x1 + 1

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U22#(mark(X1),X2) -> U22#(X1,X2)
p2: U22#(X1,active(X2)) -> U22#(X1,X2)
p3: U22#(active(X1),X2) -> U22#(X1,X2)
p4: U22#(X1,mark(X2)) -> U22#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U22#_A(x1,x2) = x2
    mark_A(x1) = x1 + 1
    active_A(x1) = x1

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U22#(mark(X1),X2) -> U22#(X1,X2)
p2: U22#(X1,active(X2)) -> U22#(X1,X2)
p3: U22#(active(X1),X2) -> U22#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U22#(mark(X1),X2) -> U22#(X1,X2)
p2: U22#(active(X1),X2) -> U22#(X1,X2)
p3: U22#(X1,active(X2)) -> U22#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U22#_A(x1,x2) = x2 + 1
    mark_A(x1) = max{0, x1 - 1}
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p3

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U22#(mark(X1),X2) -> U22#(X1,X2)
p2: U22#(active(X1),X2) -> U22#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U22#(mark(X1),X2) -> U22#(X1,X2)
p2: U22#(active(X1),X2) -> U22#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U22#_A(x1,x2) = x1
    mark_A(x1) = x1
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U22#(mark(X1),X2) -> U22#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U22#(mark(X1),X2) -> U22#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U22#_A(x1,x2) = x1
    mark_A(x1) = x1 + 1

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U23#(mark(X)) -> U23#(X)
p2: U23#(active(X)) -> U23#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U23#_A(x1) = max{0, x1 - 1}
    mark_A(x1) = x1 + 1
    active_A(x1) = x1 + 2

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U23#(mark(X)) -> U23#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U23#(mark(X)) -> U23#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  max/plus interpretations on natural numbers:
  
    U23#_A(x1) = x1
    mark_A(x1) = x1 + 1

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U32#(mark(X1),X2) -> U32#(X1,X2)
p2: U32#(X1,active(X2)) -> U32#(X1,X2)
p3: U32#(active(X1),X2) -> U32#(X1,X2)
p4: U32#(X1,mark(X2)) -> U32#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U32#_A(x1,x2) = x2
    mark_A(x1) = x1 + 1
    active_A(x1) = x1

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U32#(mark(X1),X2) -> U32#(X1,X2)
p2: U32#(X1,active(X2)) -> U32#(X1,X2)
p3: U32#(active(X1),X2) -> U32#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U32#(mark(X1),X2) -> U32#(X1,X2)
p2: U32#(active(X1),X2) -> U32#(X1,X2)
p3: U32#(X1,active(X2)) -> U32#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U32#_A(x1,x2) = x2 + 1
    mark_A(x1) = max{0, x1 - 1}
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p3

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U32#(mark(X1),X2) -> U32#(X1,X2)
p2: U32#(active(X1),X2) -> U32#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U32#(mark(X1),X2) -> U32#(X1,X2)
p2: U32#(active(X1),X2) -> U32#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U32#_A(x1,x2) = x1
    mark_A(x1) = x1
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U32#(mark(X1),X2) -> U32#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U32#(mark(X1),X2) -> U32#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U32#_A(x1,x2) = x1
    mark_A(x1) = x1 + 1

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U33#(mark(X)) -> U33#(X)
p2: U33#(active(X)) -> U33#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U33#_A(x1) = max{0, x1 - 1}
    mark_A(x1) = x1 + 1
    active_A(x1) = x1 + 2

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U33#(mark(X)) -> U33#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U33#(mark(X)) -> U33#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  max/plus interpretations on natural numbers:
  
    U33#_A(x1) = x1
    mark_A(x1) = x1 + 1

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U42#(mark(X1),X2,X3) -> U42#(X1,X2,X3)
p2: U42#(X1,X2,active(X3)) -> U42#(X1,X2,X3)
p3: U42#(X1,active(X2),X3) -> U42#(X1,X2,X3)
p4: U42#(active(X1),X2,X3) -> U42#(X1,X2,X3)
p5: U42#(X1,X2,mark(X3)) -> U42#(X1,X2,X3)
p6: U42#(X1,mark(X2),X3) -> U42#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U42#_A(x1,x2,x3) = x3 + 1
    mark_A(x1) = x1 + 1
    active_A(x1) = x1

The next rules are strictly ordered:

  p5

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U42#(mark(X1),X2,X3) -> U42#(X1,X2,X3)
p2: U42#(X1,X2,active(X3)) -> U42#(X1,X2,X3)
p3: U42#(X1,active(X2),X3) -> U42#(X1,X2,X3)
p4: U42#(active(X1),X2,X3) -> U42#(X1,X2,X3)
p5: U42#(X1,mark(X2),X3) -> U42#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U42#(mark(X1),X2,X3) -> U42#(X1,X2,X3)
p2: U42#(X1,mark(X2),X3) -> U42#(X1,X2,X3)
p3: U42#(active(X1),X2,X3) -> U42#(X1,X2,X3)
p4: U42#(X1,active(X2),X3) -> U42#(X1,X2,X3)
p5: U42#(X1,X2,active(X3)) -> U42#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U42#_A(x1,x2,x3) = x3
    mark_A(x1) = x1 + 2
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p5

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U42#(mark(X1),X2,X3) -> U42#(X1,X2,X3)
p2: U42#(X1,mark(X2),X3) -> U42#(X1,X2,X3)
p3: U42#(active(X1),X2,X3) -> U42#(X1,X2,X3)
p4: U42#(X1,active(X2),X3) -> U42#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U42#(mark(X1),X2,X3) -> U42#(X1,X2,X3)
p2: U42#(X1,active(X2),X3) -> U42#(X1,X2,X3)
p3: U42#(active(X1),X2,X3) -> U42#(X1,X2,X3)
p4: U42#(X1,mark(X2),X3) -> U42#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U42#_A(x1,x2,x3) = max{2, x2 + 1}
    mark_A(x1) = x1 + 2
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U42#(mark(X1),X2,X3) -> U42#(X1,X2,X3)
p2: U42#(X1,active(X2),X3) -> U42#(X1,X2,X3)
p3: U42#(active(X1),X2,X3) -> U42#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U42#(mark(X1),X2,X3) -> U42#(X1,X2,X3)
p2: U42#(active(X1),X2,X3) -> U42#(X1,X2,X3)
p3: U42#(X1,active(X2),X3) -> U42#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U42#_A(x1,x2,x3) = x1 + 1
    mark_A(x1) = x1
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U42#(mark(X1),X2,X3) -> U42#(X1,X2,X3)
p2: U42#(X1,active(X2),X3) -> U42#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U42#(mark(X1),X2,X3) -> U42#(X1,X2,X3)
p2: U42#(X1,active(X2),X3) -> U42#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U42#_A(x1,x2,x3) = x1
    mark_A(x1) = x1 + 1
    active_A(x1) = max{0, x1 - 1}

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U42#(X1,active(X2),X3) -> U42#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U42#(X1,active(X2),X3) -> U42#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U42#_A(x1,x2,x3) = x2
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U43#(mark(X1),X2,X3) -> U43#(X1,X2,X3)
p2: U43#(X1,X2,active(X3)) -> U43#(X1,X2,X3)
p3: U43#(X1,active(X2),X3) -> U43#(X1,X2,X3)
p4: U43#(active(X1),X2,X3) -> U43#(X1,X2,X3)
p5: U43#(X1,X2,mark(X3)) -> U43#(X1,X2,X3)
p6: U43#(X1,mark(X2),X3) -> U43#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U43#_A(x1,x2,x3) = x3 + 1
    mark_A(x1) = x1 + 1
    active_A(x1) = x1

The next rules are strictly ordered:

  p5

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U43#(mark(X1),X2,X3) -> U43#(X1,X2,X3)
p2: U43#(X1,X2,active(X3)) -> U43#(X1,X2,X3)
p3: U43#(X1,active(X2),X3) -> U43#(X1,X2,X3)
p4: U43#(active(X1),X2,X3) -> U43#(X1,X2,X3)
p5: U43#(X1,mark(X2),X3) -> U43#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U43#(mark(X1),X2,X3) -> U43#(X1,X2,X3)
p2: U43#(X1,mark(X2),X3) -> U43#(X1,X2,X3)
p3: U43#(active(X1),X2,X3) -> U43#(X1,X2,X3)
p4: U43#(X1,active(X2),X3) -> U43#(X1,X2,X3)
p5: U43#(X1,X2,active(X3)) -> U43#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U43#_A(x1,x2,x3) = x3
    mark_A(x1) = x1 + 2
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p5

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U43#(mark(X1),X2,X3) -> U43#(X1,X2,X3)
p2: U43#(X1,mark(X2),X3) -> U43#(X1,X2,X3)
p3: U43#(active(X1),X2,X3) -> U43#(X1,X2,X3)
p4: U43#(X1,active(X2),X3) -> U43#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U43#(mark(X1),X2,X3) -> U43#(X1,X2,X3)
p2: U43#(X1,active(X2),X3) -> U43#(X1,X2,X3)
p3: U43#(active(X1),X2,X3) -> U43#(X1,X2,X3)
p4: U43#(X1,mark(X2),X3) -> U43#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U43#_A(x1,x2,x3) = max{2, x2 + 1}
    mark_A(x1) = x1 + 2
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U43#(mark(X1),X2,X3) -> U43#(X1,X2,X3)
p2: U43#(X1,active(X2),X3) -> U43#(X1,X2,X3)
p3: U43#(active(X1),X2,X3) -> U43#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U43#(mark(X1),X2,X3) -> U43#(X1,X2,X3)
p2: U43#(active(X1),X2,X3) -> U43#(X1,X2,X3)
p3: U43#(X1,active(X2),X3) -> U43#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U43#_A(x1,x2,x3) = x1 + 1
    mark_A(x1) = x1
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U43#(mark(X1),X2,X3) -> U43#(X1,X2,X3)
p2: U43#(X1,active(X2),X3) -> U43#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U43#(mark(X1),X2,X3) -> U43#(X1,X2,X3)
p2: U43#(X1,active(X2),X3) -> U43#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U43#_A(x1,x2,x3) = x1
    mark_A(x1) = x1 + 1
    active_A(x1) = max{0, x1 - 1}

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U43#(X1,active(X2),X3) -> U43#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U43#(X1,active(X2),X3) -> U43#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U43#_A(x1,x2,x3) = x2
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U44#(mark(X1),X2,X3) -> U44#(X1,X2,X3)
p2: U44#(X1,X2,active(X3)) -> U44#(X1,X2,X3)
p3: U44#(X1,active(X2),X3) -> U44#(X1,X2,X3)
p4: U44#(active(X1),X2,X3) -> U44#(X1,X2,X3)
p5: U44#(X1,X2,mark(X3)) -> U44#(X1,X2,X3)
p6: U44#(X1,mark(X2),X3) -> U44#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U44#_A(x1,x2,x3) = x3 + 1
    mark_A(x1) = x1 + 1
    active_A(x1) = x1

The next rules are strictly ordered:

  p5

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U44#(mark(X1),X2,X3) -> U44#(X1,X2,X3)
p2: U44#(X1,X2,active(X3)) -> U44#(X1,X2,X3)
p3: U44#(X1,active(X2),X3) -> U44#(X1,X2,X3)
p4: U44#(active(X1),X2,X3) -> U44#(X1,X2,X3)
p5: U44#(X1,mark(X2),X3) -> U44#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U44#(mark(X1),X2,X3) -> U44#(X1,X2,X3)
p2: U44#(X1,mark(X2),X3) -> U44#(X1,X2,X3)
p3: U44#(active(X1),X2,X3) -> U44#(X1,X2,X3)
p4: U44#(X1,active(X2),X3) -> U44#(X1,X2,X3)
p5: U44#(X1,X2,active(X3)) -> U44#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U44#_A(x1,x2,x3) = x3
    mark_A(x1) = x1 + 2
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p5

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U44#(mark(X1),X2,X3) -> U44#(X1,X2,X3)
p2: U44#(X1,mark(X2),X3) -> U44#(X1,X2,X3)
p3: U44#(active(X1),X2,X3) -> U44#(X1,X2,X3)
p4: U44#(X1,active(X2),X3) -> U44#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U44#(mark(X1),X2,X3) -> U44#(X1,X2,X3)
p2: U44#(X1,active(X2),X3) -> U44#(X1,X2,X3)
p3: U44#(active(X1),X2,X3) -> U44#(X1,X2,X3)
p4: U44#(X1,mark(X2),X3) -> U44#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U44#_A(x1,x2,x3) = max{2, x2 + 1}
    mark_A(x1) = x1 + 2
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U44#(mark(X1),X2,X3) -> U44#(X1,X2,X3)
p2: U44#(X1,active(X2),X3) -> U44#(X1,X2,X3)
p3: U44#(active(X1),X2,X3) -> U44#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U44#(mark(X1),X2,X3) -> U44#(X1,X2,X3)
p2: U44#(active(X1),X2,X3) -> U44#(X1,X2,X3)
p3: U44#(X1,active(X2),X3) -> U44#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U44#_A(x1,x2,x3) = x1 + 1
    mark_A(x1) = x1
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U44#(mark(X1),X2,X3) -> U44#(X1,X2,X3)
p2: U44#(X1,active(X2),X3) -> U44#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U44#(mark(X1),X2,X3) -> U44#(X1,X2,X3)
p2: U44#(X1,active(X2),X3) -> U44#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U44#_A(x1,x2,x3) = x1
    mark_A(x1) = x1 + 1
    active_A(x1) = max{0, x1 - 1}

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U44#(X1,active(X2),X3) -> U44#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U44#(X1,active(X2),X3) -> U44#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U44#_A(x1,x2,x3) = x2
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U45#(mark(X1),X2) -> U45#(X1,X2)
p2: U45#(X1,active(X2)) -> U45#(X1,X2)
p3: U45#(active(X1),X2) -> U45#(X1,X2)
p4: U45#(X1,mark(X2)) -> U45#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U45#_A(x1,x2) = x2
    mark_A(x1) = x1 + 1
    active_A(x1) = x1

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U45#(mark(X1),X2) -> U45#(X1,X2)
p2: U45#(X1,active(X2)) -> U45#(X1,X2)
p3: U45#(active(X1),X2) -> U45#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U45#(mark(X1),X2) -> U45#(X1,X2)
p2: U45#(active(X1),X2) -> U45#(X1,X2)
p3: U45#(X1,active(X2)) -> U45#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U45#_A(x1,x2) = x2 + 1
    mark_A(x1) = max{0, x1 - 1}
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p3

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U45#(mark(X1),X2) -> U45#(X1,X2)
p2: U45#(active(X1),X2) -> U45#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U45#(mark(X1),X2) -> U45#(X1,X2)
p2: U45#(active(X1),X2) -> U45#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U45#_A(x1,x2) = x1
    mark_A(x1) = x1
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U45#(mark(X1),X2) -> U45#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U45#(mark(X1),X2) -> U45#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U45#_A(x1,x2) = x1
    mark_A(x1) = x1 + 1

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U46#(mark(X)) -> U46#(X)
p2: U46#(active(X)) -> U46#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U46#_A(x1) = max{0, x1 - 1}
    mark_A(x1) = x1 + 1
    active_A(x1) = x1 + 2

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U46#(mark(X)) -> U46#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U46#(mark(X)) -> U46#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  max/plus interpretations on natural numbers:
  
    U46#_A(x1) = x1
    mark_A(x1) = x1 + 1

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U52#(mark(X)) -> U52#(X)
p2: U52#(active(X)) -> U52#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U52#_A(x1) = max{0, x1 - 1}
    mark_A(x1) = x1 + 1
    active_A(x1) = x1 + 2

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U52#(mark(X)) -> U52#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U52#(mark(X)) -> U52#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  max/plus interpretations on natural numbers:
  
    U52#_A(x1) = x1
    mark_A(x1) = x1 + 1

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U62#(mark(X)) -> U62#(X)
p2: U62#(active(X)) -> U62#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U62#_A(x1) = max{0, x1 - 1}
    mark_A(x1) = x1 + 1
    active_A(x1) = x1 + 2

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U62#(mark(X)) -> U62#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U62#(mark(X)) -> U62#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  max/plus interpretations on natural numbers:
  
    U62#_A(x1) = x1
    mark_A(x1) = x1 + 1

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U92#(mark(X1),X2,X3) -> U92#(X1,X2,X3)
p2: U92#(X1,X2,active(X3)) -> U92#(X1,X2,X3)
p3: U92#(X1,active(X2),X3) -> U92#(X1,X2,X3)
p4: U92#(active(X1),X2,X3) -> U92#(X1,X2,X3)
p5: U92#(X1,X2,mark(X3)) -> U92#(X1,X2,X3)
p6: U92#(X1,mark(X2),X3) -> U92#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U92#_A(x1,x2,x3) = x3 + 1
    mark_A(x1) = x1 + 1
    active_A(x1) = x1

The next rules are strictly ordered:

  p5

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U92#(mark(X1),X2,X3) -> U92#(X1,X2,X3)
p2: U92#(X1,X2,active(X3)) -> U92#(X1,X2,X3)
p3: U92#(X1,active(X2),X3) -> U92#(X1,X2,X3)
p4: U92#(active(X1),X2,X3) -> U92#(X1,X2,X3)
p5: U92#(X1,mark(X2),X3) -> U92#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U92#(mark(X1),X2,X3) -> U92#(X1,X2,X3)
p2: U92#(X1,mark(X2),X3) -> U92#(X1,X2,X3)
p3: U92#(active(X1),X2,X3) -> U92#(X1,X2,X3)
p4: U92#(X1,active(X2),X3) -> U92#(X1,X2,X3)
p5: U92#(X1,X2,active(X3)) -> U92#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U92#_A(x1,x2,x3) = x3
    mark_A(x1) = x1 + 2
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p5

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U92#(mark(X1),X2,X3) -> U92#(X1,X2,X3)
p2: U92#(X1,mark(X2),X3) -> U92#(X1,X2,X3)
p3: U92#(active(X1),X2,X3) -> U92#(X1,X2,X3)
p4: U92#(X1,active(X2),X3) -> U92#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U92#(mark(X1),X2,X3) -> U92#(X1,X2,X3)
p2: U92#(X1,active(X2),X3) -> U92#(X1,X2,X3)
p3: U92#(active(X1),X2,X3) -> U92#(X1,X2,X3)
p4: U92#(X1,mark(X2),X3) -> U92#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U92#_A(x1,x2,x3) = max{0, x2 - 1}
    mark_A(x1) = x1 + 2
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U92#(mark(X1),X2,X3) -> U92#(X1,X2,X3)
p2: U92#(X1,active(X2),X3) -> U92#(X1,X2,X3)
p3: U92#(active(X1),X2,X3) -> U92#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U92#(mark(X1),X2,X3) -> U92#(X1,X2,X3)
p2: U92#(active(X1),X2,X3) -> U92#(X1,X2,X3)
p3: U92#(X1,active(X2),X3) -> U92#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U92#_A(x1,x2,x3) = x1 + 1
    mark_A(x1) = x1
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U92#(mark(X1),X2,X3) -> U92#(X1,X2,X3)
p2: U92#(X1,active(X2),X3) -> U92#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U92#(mark(X1),X2,X3) -> U92#(X1,X2,X3)
p2: U92#(X1,active(X2),X3) -> U92#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U92#_A(x1,x2,x3) = x1
    mark_A(x1) = x1 + 1
    active_A(x1) = max{0, x1 - 1}

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U92#(X1,active(X2),X3) -> U92#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U92#(X1,active(X2),X3) -> U92#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U92#_A(x1,x2,x3) = x2
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U93#(mark(X1),X2,X3) -> U93#(X1,X2,X3)
p2: U93#(X1,X2,active(X3)) -> U93#(X1,X2,X3)
p3: U93#(X1,active(X2),X3) -> U93#(X1,X2,X3)
p4: U93#(active(X1),X2,X3) -> U93#(X1,X2,X3)
p5: U93#(X1,X2,mark(X3)) -> U93#(X1,X2,X3)
p6: U93#(X1,mark(X2),X3) -> U93#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U93#_A(x1,x2,x3) = x3 + 1
    mark_A(x1) = x1 + 1
    active_A(x1) = x1

The next rules are strictly ordered:

  p5

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U93#(mark(X1),X2,X3) -> U93#(X1,X2,X3)
p2: U93#(X1,X2,active(X3)) -> U93#(X1,X2,X3)
p3: U93#(X1,active(X2),X3) -> U93#(X1,X2,X3)
p4: U93#(active(X1),X2,X3) -> U93#(X1,X2,X3)
p5: U93#(X1,mark(X2),X3) -> U93#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U93#(mark(X1),X2,X3) -> U93#(X1,X2,X3)
p2: U93#(X1,mark(X2),X3) -> U93#(X1,X2,X3)
p3: U93#(active(X1),X2,X3) -> U93#(X1,X2,X3)
p4: U93#(X1,active(X2),X3) -> U93#(X1,X2,X3)
p5: U93#(X1,X2,active(X3)) -> U93#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U93#_A(x1,x2,x3) = x3
    mark_A(x1) = x1 + 2
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p5

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U93#(mark(X1),X2,X3) -> U93#(X1,X2,X3)
p2: U93#(X1,mark(X2),X3) -> U93#(X1,X2,X3)
p3: U93#(active(X1),X2,X3) -> U93#(X1,X2,X3)
p4: U93#(X1,active(X2),X3) -> U93#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U93#(mark(X1),X2,X3) -> U93#(X1,X2,X3)
p2: U93#(X1,active(X2),X3) -> U93#(X1,X2,X3)
p3: U93#(active(X1),X2,X3) -> U93#(X1,X2,X3)
p4: U93#(X1,mark(X2),X3) -> U93#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U93#_A(x1,x2,x3) = max{2, x2 + 1}
    mark_A(x1) = x1 + 2
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U93#(mark(X1),X2,X3) -> U93#(X1,X2,X3)
p2: U93#(X1,active(X2),X3) -> U93#(X1,X2,X3)
p3: U93#(active(X1),X2,X3) -> U93#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U93#(mark(X1),X2,X3) -> U93#(X1,X2,X3)
p2: U93#(active(X1),X2,X3) -> U93#(X1,X2,X3)
p3: U93#(X1,active(X2),X3) -> U93#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U93#_A(x1,x2,x3) = x1 + 1
    mark_A(x1) = x1
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U93#(mark(X1),X2,X3) -> U93#(X1,X2,X3)
p2: U93#(X1,active(X2),X3) -> U93#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U93#(mark(X1),X2,X3) -> U93#(X1,X2,X3)
p2: U93#(X1,active(X2),X3) -> U93#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U93#_A(x1,x2,x3) = x1
    mark_A(x1) = x1 + 1
    active_A(x1) = max{0, x1 - 1}

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U93#(X1,active(X2),X3) -> U93#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U93#(X1,active(X2),X3) -> U93#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U93#_A(x1,x2,x3) = x2
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U94#(mark(X1),X2,X3) -> U94#(X1,X2,X3)
p2: U94#(X1,X2,active(X3)) -> U94#(X1,X2,X3)
p3: U94#(X1,active(X2),X3) -> U94#(X1,X2,X3)
p4: U94#(active(X1),X2,X3) -> U94#(X1,X2,X3)
p5: U94#(X1,X2,mark(X3)) -> U94#(X1,X2,X3)
p6: U94#(X1,mark(X2),X3) -> U94#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U94#_A(x1,x2,x3) = x3 + 1
    mark_A(x1) = x1 + 1
    active_A(x1) = x1

The next rules are strictly ordered:

  p5

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U94#(mark(X1),X2,X3) -> U94#(X1,X2,X3)
p2: U94#(X1,X2,active(X3)) -> U94#(X1,X2,X3)
p3: U94#(X1,active(X2),X3) -> U94#(X1,X2,X3)
p4: U94#(active(X1),X2,X3) -> U94#(X1,X2,X3)
p5: U94#(X1,mark(X2),X3) -> U94#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U94#(mark(X1),X2,X3) -> U94#(X1,X2,X3)
p2: U94#(X1,mark(X2),X3) -> U94#(X1,X2,X3)
p3: U94#(active(X1),X2,X3) -> U94#(X1,X2,X3)
p4: U94#(X1,active(X2),X3) -> U94#(X1,X2,X3)
p5: U94#(X1,X2,active(X3)) -> U94#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U94#_A(x1,x2,x3) = x3
    mark_A(x1) = x1 + 2
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p5

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U94#(mark(X1),X2,X3) -> U94#(X1,X2,X3)
p2: U94#(X1,mark(X2),X3) -> U94#(X1,X2,X3)
p3: U94#(active(X1),X2,X3) -> U94#(X1,X2,X3)
p4: U94#(X1,active(X2),X3) -> U94#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U94#(mark(X1),X2,X3) -> U94#(X1,X2,X3)
p2: U94#(X1,active(X2),X3) -> U94#(X1,X2,X3)
p3: U94#(active(X1),X2,X3) -> U94#(X1,X2,X3)
p4: U94#(X1,mark(X2),X3) -> U94#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U94#_A(x1,x2,x3) = max{2, x2 + 1}
    mark_A(x1) = x1 + 2
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U94#(mark(X1),X2,X3) -> U94#(X1,X2,X3)
p2: U94#(X1,active(X2),X3) -> U94#(X1,X2,X3)
p3: U94#(active(X1),X2,X3) -> U94#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U94#(mark(X1),X2,X3) -> U94#(X1,X2,X3)
p2: U94#(active(X1),X2,X3) -> U94#(X1,X2,X3)
p3: U94#(X1,active(X2),X3) -> U94#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U94#_A(x1,x2,x3) = x1 + 1
    mark_A(x1) = x1
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U94#(mark(X1),X2,X3) -> U94#(X1,X2,X3)
p2: U94#(X1,active(X2),X3) -> U94#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U94#(mark(X1),X2,X3) -> U94#(X1,X2,X3)
p2: U94#(X1,active(X2),X3) -> U94#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U94#_A(x1,x2,x3) = x1
    mark_A(x1) = x1 + 1
    active_A(x1) = max{0, x1 - 1}

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U94#(X1,active(X2),X3) -> U94#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U94#(X1,active(X2),X3) -> U94#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U94#_A(x1,x2,x3) = x2
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U95#(mark(X1),X2) -> U95#(X1,X2)
p2: U95#(X1,active(X2)) -> U95#(X1,X2)
p3: U95#(active(X1),X2) -> U95#(X1,X2)
p4: U95#(X1,mark(X2)) -> U95#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U95#_A(x1,x2) = x2
    mark_A(x1) = x1 + 1
    active_A(x1) = x1

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U95#(mark(X1),X2) -> U95#(X1,X2)
p2: U95#(X1,active(X2)) -> U95#(X1,X2)
p3: U95#(active(X1),X2) -> U95#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U95#(mark(X1),X2) -> U95#(X1,X2)
p2: U95#(active(X1),X2) -> U95#(X1,X2)
p3: U95#(X1,active(X2)) -> U95#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U95#_A(x1,x2) = x2 + 1
    mark_A(x1) = max{0, x1 - 1}
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p3

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U95#(mark(X1),X2) -> U95#(X1,X2)
p2: U95#(active(X1),X2) -> U95#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U95#(mark(X1),X2) -> U95#(X1,X2)
p2: U95#(active(X1),X2) -> U95#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U95#_A(x1,x2) = x1
    mark_A(x1) = x1
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U95#(mark(X1),X2) -> U95#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U95#(mark(X1),X2) -> U95#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U95#_A(x1,x2) = x1
    mark_A(x1) = x1 + 1

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U96#(mark(X)) -> U96#(X)
p2: U96#(active(X)) -> U96#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U96#_A(x1) = max{0, x1 - 1}
    mark_A(x1) = x1 + 1
    active_A(x1) = x1 + 2

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U96#(mark(X)) -> U96#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U96#(mark(X)) -> U96#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  max/plus interpretations on natural numbers:
  
    U96#_A(x1) = x1
    mark_A(x1) = x1 + 1

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U11#(mark(X1),X2) -> U11#(X1,X2)
p2: U11#(X1,active(X2)) -> U11#(X1,X2)
p3: U11#(active(X1),X2) -> U11#(X1,X2)
p4: U11#(X1,mark(X2)) -> U11#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U11#_A(x1,x2) = x2
    mark_A(x1) = x1 + 1
    active_A(x1) = x1

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U11#(mark(X1),X2) -> U11#(X1,X2)
p2: U11#(X1,active(X2)) -> U11#(X1,X2)
p3: U11#(active(X1),X2) -> U11#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U11#(mark(X1),X2) -> U11#(X1,X2)
p2: U11#(active(X1),X2) -> U11#(X1,X2)
p3: U11#(X1,active(X2)) -> U11#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U11#_A(x1,x2) = x2 + 1
    mark_A(x1) = max{0, x1 - 1}
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p3

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U11#(mark(X1),X2) -> U11#(X1,X2)
p2: U11#(active(X1),X2) -> U11#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U11#(mark(X1),X2) -> U11#(X1,X2)
p2: U11#(active(X1),X2) -> U11#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U11#_A(x1,x2) = x1
    mark_A(x1) = x1
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U11#(mark(X1),X2) -> U11#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U11#(mark(X1),X2) -> U11#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U11#_A(x1,x2) = x1
    mark_A(x1) = x1 + 1

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U21#(mark(X1),X2) -> U21#(X1,X2)
p2: U21#(X1,active(X2)) -> U21#(X1,X2)
p3: U21#(active(X1),X2) -> U21#(X1,X2)
p4: U21#(X1,mark(X2)) -> U21#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U21#_A(x1,x2) = x2
    mark_A(x1) = x1 + 1
    active_A(x1) = x1

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U21#(mark(X1),X2) -> U21#(X1,X2)
p2: U21#(X1,active(X2)) -> U21#(X1,X2)
p3: U21#(active(X1),X2) -> U21#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U21#(mark(X1),X2) -> U21#(X1,X2)
p2: U21#(active(X1),X2) -> U21#(X1,X2)
p3: U21#(X1,active(X2)) -> U21#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U21#_A(x1,x2) = x2 + 1
    mark_A(x1) = max{0, x1 - 1}
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p3

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U21#(mark(X1),X2) -> U21#(X1,X2)
p2: U21#(active(X1),X2) -> U21#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U21#(mark(X1),X2) -> U21#(X1,X2)
p2: U21#(active(X1),X2) -> U21#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U21#_A(x1,x2) = x1
    mark_A(x1) = x1
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U21#(mark(X1),X2) -> U21#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U21#(mark(X1),X2) -> U21#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U21#_A(x1,x2) = x1
    mark_A(x1) = x1 + 1

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U31#(mark(X1),X2) -> U31#(X1,X2)
p2: U31#(X1,active(X2)) -> U31#(X1,X2)
p3: U31#(active(X1),X2) -> U31#(X1,X2)
p4: U31#(X1,mark(X2)) -> U31#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U31#_A(x1,x2) = x2
    mark_A(x1) = x1 + 1
    active_A(x1) = x1

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U31#(mark(X1),X2) -> U31#(X1,X2)
p2: U31#(X1,active(X2)) -> U31#(X1,X2)
p3: U31#(active(X1),X2) -> U31#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U31#(mark(X1),X2) -> U31#(X1,X2)
p2: U31#(active(X1),X2) -> U31#(X1,X2)
p3: U31#(X1,active(X2)) -> U31#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U31#_A(x1,x2) = x2 + 1
    mark_A(x1) = max{0, x1 - 1}
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p3

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U31#(mark(X1),X2) -> U31#(X1,X2)
p2: U31#(active(X1),X2) -> U31#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U31#(mark(X1),X2) -> U31#(X1,X2)
p2: U31#(active(X1),X2) -> U31#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U31#_A(x1,x2) = x1
    mark_A(x1) = x1
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U31#(mark(X1),X2) -> U31#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U31#(mark(X1),X2) -> U31#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U31#_A(x1,x2) = x1
    mark_A(x1) = x1 + 1

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U41#(mark(X1),X2,X3) -> U41#(X1,X2,X3)
p2: U41#(X1,X2,active(X3)) -> U41#(X1,X2,X3)
p3: U41#(X1,active(X2),X3) -> U41#(X1,X2,X3)
p4: U41#(active(X1),X2,X3) -> U41#(X1,X2,X3)
p5: U41#(X1,X2,mark(X3)) -> U41#(X1,X2,X3)
p6: U41#(X1,mark(X2),X3) -> U41#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U41#_A(x1,x2,x3) = x3 + 1
    mark_A(x1) = x1 + 1
    active_A(x1) = x1

The next rules are strictly ordered:

  p5

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U41#(mark(X1),X2,X3) -> U41#(X1,X2,X3)
p2: U41#(X1,X2,active(X3)) -> U41#(X1,X2,X3)
p3: U41#(X1,active(X2),X3) -> U41#(X1,X2,X3)
p4: U41#(active(X1),X2,X3) -> U41#(X1,X2,X3)
p5: U41#(X1,mark(X2),X3) -> U41#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U41#(mark(X1),X2,X3) -> U41#(X1,X2,X3)
p2: U41#(X1,mark(X2),X3) -> U41#(X1,X2,X3)
p3: U41#(active(X1),X2,X3) -> U41#(X1,X2,X3)
p4: U41#(X1,active(X2),X3) -> U41#(X1,X2,X3)
p5: U41#(X1,X2,active(X3)) -> U41#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U41#_A(x1,x2,x3) = x3
    mark_A(x1) = x1 + 2
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p5

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U41#(mark(X1),X2,X3) -> U41#(X1,X2,X3)
p2: U41#(X1,mark(X2),X3) -> U41#(X1,X2,X3)
p3: U41#(active(X1),X2,X3) -> U41#(X1,X2,X3)
p4: U41#(X1,active(X2),X3) -> U41#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U41#(mark(X1),X2,X3) -> U41#(X1,X2,X3)
p2: U41#(X1,active(X2),X3) -> U41#(X1,X2,X3)
p3: U41#(active(X1),X2,X3) -> U41#(X1,X2,X3)
p4: U41#(X1,mark(X2),X3) -> U41#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U41#_A(x1,x2,x3) = max{0, x2 - 1}
    mark_A(x1) = x1 + 2
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U41#(mark(X1),X2,X3) -> U41#(X1,X2,X3)
p2: U41#(X1,active(X2),X3) -> U41#(X1,X2,X3)
p3: U41#(active(X1),X2,X3) -> U41#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U41#(mark(X1),X2,X3) -> U41#(X1,X2,X3)
p2: U41#(active(X1),X2,X3) -> U41#(X1,X2,X3)
p3: U41#(X1,active(X2),X3) -> U41#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U41#_A(x1,x2,x3) = x1 + 1
    mark_A(x1) = x1
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U41#(mark(X1),X2,X3) -> U41#(X1,X2,X3)
p2: U41#(X1,active(X2),X3) -> U41#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U41#(mark(X1),X2,X3) -> U41#(X1,X2,X3)
p2: U41#(X1,active(X2),X3) -> U41#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U41#_A(x1,x2,x3) = x1
    mark_A(x1) = x1 + 1
    active_A(x1) = max{0, x1 - 1}

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U41#(X1,active(X2),X3) -> U41#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U41#(X1,active(X2),X3) -> U41#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U41#_A(x1,x2,x3) = x2
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U51#(mark(X1),X2) -> U51#(X1,X2)
p2: U51#(X1,active(X2)) -> U51#(X1,X2)
p3: U51#(active(X1),X2) -> U51#(X1,X2)
p4: U51#(X1,mark(X2)) -> U51#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U51#_A(x1,x2) = x2
    mark_A(x1) = x1 + 1
    active_A(x1) = x1

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U51#(mark(X1),X2) -> U51#(X1,X2)
p2: U51#(X1,active(X2)) -> U51#(X1,X2)
p3: U51#(active(X1),X2) -> U51#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U51#(mark(X1),X2) -> U51#(X1,X2)
p2: U51#(active(X1),X2) -> U51#(X1,X2)
p3: U51#(X1,active(X2)) -> U51#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U51#_A(x1,x2) = x2 + 1
    mark_A(x1) = max{0, x1 - 1}
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p3

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U51#(mark(X1),X2) -> U51#(X1,X2)
p2: U51#(active(X1),X2) -> U51#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U51#(mark(X1),X2) -> U51#(X1,X2)
p2: U51#(active(X1),X2) -> U51#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U51#_A(x1,x2) = x1
    mark_A(x1) = x1
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U51#(mark(X1),X2) -> U51#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U51#(mark(X1),X2) -> U51#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U51#_A(x1,x2) = x1
    mark_A(x1) = x1 + 1

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U61#(mark(X1),X2) -> U61#(X1,X2)
p2: U61#(X1,active(X2)) -> U61#(X1,X2)
p3: U61#(active(X1),X2) -> U61#(X1,X2)
p4: U61#(X1,mark(X2)) -> U61#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U61#_A(x1,x2) = x2
    mark_A(x1) = x1 + 1
    active_A(x1) = x1

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U61#(mark(X1),X2) -> U61#(X1,X2)
p2: U61#(X1,active(X2)) -> U61#(X1,X2)
p3: U61#(active(X1),X2) -> U61#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U61#(mark(X1),X2) -> U61#(X1,X2)
p2: U61#(active(X1),X2) -> U61#(X1,X2)
p3: U61#(X1,active(X2)) -> U61#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U61#_A(x1,x2) = x2 + 1
    mark_A(x1) = max{0, x1 - 1}
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p3

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U61#(mark(X1),X2) -> U61#(X1,X2)
p2: U61#(active(X1),X2) -> U61#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U61#(mark(X1),X2) -> U61#(X1,X2)
p2: U61#(active(X1),X2) -> U61#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U61#_A(x1,x2) = x1
    mark_A(x1) = x1
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U61#(mark(X1),X2) -> U61#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U61#(mark(X1),X2) -> U61#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U61#_A(x1,x2) = x1
    mark_A(x1) = x1 + 1

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U71#(mark(X)) -> U71#(X)
p2: U71#(active(X)) -> U71#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U71#_A(x1) = max{0, x1 - 1}
    mark_A(x1) = x1 + 1
    active_A(x1) = x1 + 2

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U71#(mark(X)) -> U71#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U71#(mark(X)) -> U71#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  max/plus interpretations on natural numbers:
  
    U71#_A(x1) = x1
    mark_A(x1) = x1 + 1

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U81#(mark(X)) -> U81#(X)
p2: U81#(active(X)) -> U81#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U81#_A(x1) = max{0, x1 - 1}
    mark_A(x1) = x1 + 1
    active_A(x1) = x1 + 2

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U81#(mark(X)) -> U81#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U81#(mark(X)) -> U81#(X)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the monotone reduction pair:

  max/plus interpretations on natural numbers:
  
    U81#_A(x1) = x1
    mark_A(x1) = x1 + 1

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U91#(mark(X1),X2,X3) -> U91#(X1,X2,X3)
p2: U91#(X1,X2,active(X3)) -> U91#(X1,X2,X3)
p3: U91#(X1,active(X2),X3) -> U91#(X1,X2,X3)
p4: U91#(active(X1),X2,X3) -> U91#(X1,X2,X3)
p5: U91#(X1,X2,mark(X3)) -> U91#(X1,X2,X3)
p6: U91#(X1,mark(X2),X3) -> U91#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U91#_A(x1,x2,x3) = x3 + 1
    mark_A(x1) = x1 + 1
    active_A(x1) = x1

The next rules are strictly ordered:

  p5

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U91#(mark(X1),X2,X3) -> U91#(X1,X2,X3)
p2: U91#(X1,X2,active(X3)) -> U91#(X1,X2,X3)
p3: U91#(X1,active(X2),X3) -> U91#(X1,X2,X3)
p4: U91#(active(X1),X2,X3) -> U91#(X1,X2,X3)
p5: U91#(X1,mark(X2),X3) -> U91#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U91#(mark(X1),X2,X3) -> U91#(X1,X2,X3)
p2: U91#(X1,mark(X2),X3) -> U91#(X1,X2,X3)
p3: U91#(active(X1),X2,X3) -> U91#(X1,X2,X3)
p4: U91#(X1,active(X2),X3) -> U91#(X1,X2,X3)
p5: U91#(X1,X2,active(X3)) -> U91#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U91#_A(x1,x2,x3) = x3
    mark_A(x1) = x1 + 2
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p5

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U91#(mark(X1),X2,X3) -> U91#(X1,X2,X3)
p2: U91#(X1,mark(X2),X3) -> U91#(X1,X2,X3)
p3: U91#(active(X1),X2,X3) -> U91#(X1,X2,X3)
p4: U91#(X1,active(X2),X3) -> U91#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U91#(mark(X1),X2,X3) -> U91#(X1,X2,X3)
p2: U91#(X1,active(X2),X3) -> U91#(X1,X2,X3)
p3: U91#(active(X1),X2,X3) -> U91#(X1,X2,X3)
p4: U91#(X1,mark(X2),X3) -> U91#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U91#_A(x1,x2,x3) = max{2, x2 + 1}
    mark_A(x1) = x1 + 2
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U91#(mark(X1),X2,X3) -> U91#(X1,X2,X3)
p2: U91#(X1,active(X2),X3) -> U91#(X1,X2,X3)
p3: U91#(active(X1),X2,X3) -> U91#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U91#(mark(X1),X2,X3) -> U91#(X1,X2,X3)
p2: U91#(active(X1),X2,X3) -> U91#(X1,X2,X3)
p3: U91#(X1,active(X2),X3) -> U91#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U91#_A(x1,x2,x3) = x1 + 1
    mark_A(x1) = x1
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U91#(mark(X1),X2,X3) -> U91#(X1,X2,X3)
p2: U91#(X1,active(X2),X3) -> U91#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U91#(mark(X1),X2,X3) -> U91#(X1,X2,X3)
p2: U91#(X1,active(X2),X3) -> U91#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U91#_A(x1,x2,x3) = x1
    mark_A(x1) = x1 + 1
    active_A(x1) = max{0, x1 - 1}

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U91#(X1,active(X2),X3) -> U91#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U91#(X1,active(X2),X3) -> U91#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U91#_A(x1,x2,x3) = x2
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U101#(mark(X1),X2,X3) -> U101#(X1,X2,X3)
p2: U101#(X1,X2,active(X3)) -> U101#(X1,X2,X3)
p3: U101#(X1,active(X2),X3) -> U101#(X1,X2,X3)
p4: U101#(active(X1),X2,X3) -> U101#(X1,X2,X3)
p5: U101#(X1,X2,mark(X3)) -> U101#(X1,X2,X3)
p6: U101#(X1,mark(X2),X3) -> U101#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U101#_A(x1,x2,x3) = x3 + 1
    mark_A(x1) = x1 + 1
    active_A(x1) = x1

The next rules are strictly ordered:

  p5

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U101#(mark(X1),X2,X3) -> U101#(X1,X2,X3)
p2: U101#(X1,X2,active(X3)) -> U101#(X1,X2,X3)
p3: U101#(X1,active(X2),X3) -> U101#(X1,X2,X3)
p4: U101#(active(X1),X2,X3) -> U101#(X1,X2,X3)
p5: U101#(X1,mark(X2),X3) -> U101#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U101#(mark(X1),X2,X3) -> U101#(X1,X2,X3)
p2: U101#(X1,mark(X2),X3) -> U101#(X1,X2,X3)
p3: U101#(active(X1),X2,X3) -> U101#(X1,X2,X3)
p4: U101#(X1,active(X2),X3) -> U101#(X1,X2,X3)
p5: U101#(X1,X2,active(X3)) -> U101#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U101#_A(x1,x2,x3) = x3
    mark_A(x1) = x1 + 2
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p5

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U101#(mark(X1),X2,X3) -> U101#(X1,X2,X3)
p2: U101#(X1,mark(X2),X3) -> U101#(X1,X2,X3)
p3: U101#(active(X1),X2,X3) -> U101#(X1,X2,X3)
p4: U101#(X1,active(X2),X3) -> U101#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U101#(mark(X1),X2,X3) -> U101#(X1,X2,X3)
p2: U101#(X1,active(X2),X3) -> U101#(X1,X2,X3)
p3: U101#(active(X1),X2,X3) -> U101#(X1,X2,X3)
p4: U101#(X1,mark(X2),X3) -> U101#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U101#_A(x1,x2,x3) = max{0, x2 - 1}
    mark_A(x1) = x1 + 2
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U101#(mark(X1),X2,X3) -> U101#(X1,X2,X3)
p2: U101#(X1,active(X2),X3) -> U101#(X1,X2,X3)
p3: U101#(active(X1),X2,X3) -> U101#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U101#(mark(X1),X2,X3) -> U101#(X1,X2,X3)
p2: U101#(active(X1),X2,X3) -> U101#(X1,X2,X3)
p3: U101#(X1,active(X2),X3) -> U101#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U101#_A(x1,x2,x3) = x1 + 1
    mark_A(x1) = x1
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U101#(mark(X1),X2,X3) -> U101#(X1,X2,X3)
p2: U101#(X1,active(X2),X3) -> U101#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U101#(mark(X1),X2,X3) -> U101#(X1,X2,X3)
p2: U101#(X1,active(X2),X3) -> U101#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U101#_A(x1,x2,x3) = x1
    mark_A(x1) = x1 + 1
    active_A(x1) = max{0, x1 - 1}

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U101#(X1,active(X2),X3) -> U101#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U101#(X1,active(X2),X3) -> U101#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U101#_A(x1,x2,x3) = x2
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U111#(mark(X1),X2,X3) -> U111#(X1,X2,X3)
p2: U111#(X1,X2,active(X3)) -> U111#(X1,X2,X3)
p3: U111#(X1,active(X2),X3) -> U111#(X1,X2,X3)
p4: U111#(active(X1),X2,X3) -> U111#(X1,X2,X3)
p5: U111#(X1,X2,mark(X3)) -> U111#(X1,X2,X3)
p6: U111#(X1,mark(X2),X3) -> U111#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U111#_A(x1,x2,x3) = x3 + 1
    mark_A(x1) = x1 + 1
    active_A(x1) = x1

The next rules are strictly ordered:

  p5

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U111#(mark(X1),X2,X3) -> U111#(X1,X2,X3)
p2: U111#(X1,X2,active(X3)) -> U111#(X1,X2,X3)
p3: U111#(X1,active(X2),X3) -> U111#(X1,X2,X3)
p4: U111#(active(X1),X2,X3) -> U111#(X1,X2,X3)
p5: U111#(X1,mark(X2),X3) -> U111#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U111#(mark(X1),X2,X3) -> U111#(X1,X2,X3)
p2: U111#(X1,mark(X2),X3) -> U111#(X1,X2,X3)
p3: U111#(active(X1),X2,X3) -> U111#(X1,X2,X3)
p4: U111#(X1,active(X2),X3) -> U111#(X1,X2,X3)
p5: U111#(X1,X2,active(X3)) -> U111#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U111#_A(x1,x2,x3) = x3
    mark_A(x1) = x1 + 2
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p5

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U111#(mark(X1),X2,X3) -> U111#(X1,X2,X3)
p2: U111#(X1,mark(X2),X3) -> U111#(X1,X2,X3)
p3: U111#(active(X1),X2,X3) -> U111#(X1,X2,X3)
p4: U111#(X1,active(X2),X3) -> U111#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U111#(mark(X1),X2,X3) -> U111#(X1,X2,X3)
p2: U111#(X1,active(X2),X3) -> U111#(X1,X2,X3)
p3: U111#(active(X1),X2,X3) -> U111#(X1,X2,X3)
p4: U111#(X1,mark(X2),X3) -> U111#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U111#_A(x1,x2,x3) = max{0, x2 - 1}
    mark_A(x1) = x1 + 2
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U111#(mark(X1),X2,X3) -> U111#(X1,X2,X3)
p2: U111#(X1,active(X2),X3) -> U111#(X1,X2,X3)
p3: U111#(active(X1),X2,X3) -> U111#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U111#(mark(X1),X2,X3) -> U111#(X1,X2,X3)
p2: U111#(active(X1),X2,X3) -> U111#(X1,X2,X3)
p3: U111#(X1,active(X2),X3) -> U111#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U111#_A(x1,x2,x3) = x1 + 1
    mark_A(x1) = x1
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U111#(mark(X1),X2,X3) -> U111#(X1,X2,X3)
p2: U111#(X1,active(X2),X3) -> U111#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U111#(mark(X1),X2,X3) -> U111#(X1,X2,X3)
p2: U111#(X1,active(X2),X3) -> U111#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U111#_A(x1,x2,x3) = x1
    mark_A(x1) = x1 + 1
    active_A(x1) = max{0, x1 - 1}

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U111#(X1,active(X2),X3) -> U111#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U111#(X1,active(X2),X3) -> U111#(X1,X2,X3)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U111#_A(x1,x2,x3) = x2
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U121#(mark(X1),X2) -> U121#(X1,X2)
p2: U121#(X1,active(X2)) -> U121#(X1,X2)
p3: U121#(active(X1),X2) -> U121#(X1,X2)
p4: U121#(X1,mark(X2)) -> U121#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U121#_A(x1,x2) = x2
    mark_A(x1) = x1 + 1
    active_A(x1) = x1

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U121#(mark(X1),X2) -> U121#(X1,X2)
p2: U121#(X1,active(X2)) -> U121#(X1,X2)
p3: U121#(active(X1),X2) -> U121#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U121#(mark(X1),X2) -> U121#(X1,X2)
p2: U121#(active(X1),X2) -> U121#(X1,X2)
p3: U121#(X1,active(X2)) -> U121#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U121#_A(x1,x2) = x2 + 1
    mark_A(x1) = max{0, x1 - 1}
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p3

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U121#(mark(X1),X2) -> U121#(X1,X2)
p2: U121#(active(X1),X2) -> U121#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U121#(mark(X1),X2) -> U121#(X1,X2)
p2: U121#(active(X1),X2) -> U121#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U121#_A(x1,x2) = x1
    mark_A(x1) = x1
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U121#(mark(X1),X2) -> U121#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U121#(mark(X1),X2) -> U121#(X1,X2)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U121#_A(x1,x2) = x1
    mark_A(x1) = x1 + 1

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.

-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U131#(mark(X1),X2,X3,X4) -> U131#(X1,X2,X3,X4)
p2: U131#(X1,X2,X3,active(X4)) -> U131#(X1,X2,X3,X4)
p3: U131#(X1,X2,active(X3),X4) -> U131#(X1,X2,X3,X4)
p4: U131#(X1,active(X2),X3,X4) -> U131#(X1,X2,X3,X4)
p5: U131#(active(X1),X2,X3,X4) -> U131#(X1,X2,X3,X4)
p6: U131#(X1,X2,X3,mark(X4)) -> U131#(X1,X2,X3,X4)
p7: U131#(X1,X2,mark(X3),X4) -> U131#(X1,X2,X3,X4)
p8: U131#(X1,mark(X2),X3,X4) -> U131#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U131#_A(x1,x2,x3,x4) = max{0, x4 - 1}
    mark_A(x1) = x1 + 2
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p6

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U131#(mark(X1),X2,X3,X4) -> U131#(X1,X2,X3,X4)
p2: U131#(X1,X2,X3,active(X4)) -> U131#(X1,X2,X3,X4)
p3: U131#(X1,X2,active(X3),X4) -> U131#(X1,X2,X3,X4)
p4: U131#(X1,active(X2),X3,X4) -> U131#(X1,X2,X3,X4)
p5: U131#(active(X1),X2,X3,X4) -> U131#(X1,X2,X3,X4)
p6: U131#(X1,X2,mark(X3),X4) -> U131#(X1,X2,X3,X4)
p7: U131#(X1,mark(X2),X3,X4) -> U131#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6, p7}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U131#(mark(X1),X2,X3,X4) -> U131#(X1,X2,X3,X4)
p2: U131#(X1,mark(X2),X3,X4) -> U131#(X1,X2,X3,X4)
p3: U131#(X1,X2,mark(X3),X4) -> U131#(X1,X2,X3,X4)
p4: U131#(active(X1),X2,X3,X4) -> U131#(X1,X2,X3,X4)
p5: U131#(X1,active(X2),X3,X4) -> U131#(X1,X2,X3,X4)
p6: U131#(X1,X2,active(X3),X4) -> U131#(X1,X2,X3,X4)
p7: U131#(X1,X2,X3,active(X4)) -> U131#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U131#_A(x1,x2,x3,x4) = x1 + 1
    mark_A(x1) = x1
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U131#(mark(X1),X2,X3,X4) -> U131#(X1,X2,X3,X4)
p2: U131#(X1,mark(X2),X3,X4) -> U131#(X1,X2,X3,X4)
p3: U131#(X1,X2,mark(X3),X4) -> U131#(X1,X2,X3,X4)
p4: U131#(X1,active(X2),X3,X4) -> U131#(X1,X2,X3,X4)
p5: U131#(X1,X2,active(X3),X4) -> U131#(X1,X2,X3,X4)
p6: U131#(X1,X2,X3,active(X4)) -> U131#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5, p6}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U131#(mark(X1),X2,X3,X4) -> U131#(X1,X2,X3,X4)
p2: U131#(X1,X2,X3,active(X4)) -> U131#(X1,X2,X3,X4)
p3: U131#(X1,X2,active(X3),X4) -> U131#(X1,X2,X3,X4)
p4: U131#(X1,active(X2),X3,X4) -> U131#(X1,X2,X3,X4)
p5: U131#(X1,X2,mark(X3),X4) -> U131#(X1,X2,X3,X4)
p6: U131#(X1,mark(X2),X3,X4) -> U131#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U131#_A(x1,x2,x3,x4) = x1 + 3
    mark_A(x1) = x1 + 1
    active_A(x1) = max{2, x1 - 1}

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U131#(X1,X2,X3,active(X4)) -> U131#(X1,X2,X3,X4)
p2: U131#(X1,X2,active(X3),X4) -> U131#(X1,X2,X3,X4)
p3: U131#(X1,active(X2),X3,X4) -> U131#(X1,X2,X3,X4)
p4: U131#(X1,X2,mark(X3),X4) -> U131#(X1,X2,X3,X4)
p5: U131#(X1,mark(X2),X3,X4) -> U131#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4, p5}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U131#(X1,X2,X3,active(X4)) -> U131#(X1,X2,X3,X4)
p2: U131#(X1,mark(X2),X3,X4) -> U131#(X1,X2,X3,X4)
p3: U131#(X1,X2,mark(X3),X4) -> U131#(X1,X2,X3,X4)
p4: U131#(X1,active(X2),X3,X4) -> U131#(X1,X2,X3,X4)
p5: U131#(X1,X2,active(X3),X4) -> U131#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U131#_A(x1,x2,x3,x4) = x4
    active_A(x1) = x1 + 1
    mark_A(x1) = max{0, x1 - 1}

The next rules are strictly ordered:

  p1

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U131#(X1,mark(X2),X3,X4) -> U131#(X1,X2,X3,X4)
p2: U131#(X1,X2,mark(X3),X4) -> U131#(X1,X2,X3,X4)
p3: U131#(X1,active(X2),X3,X4) -> U131#(X1,X2,X3,X4)
p4: U131#(X1,X2,active(X3),X4) -> U131#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3, p4}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U131#(X1,mark(X2),X3,X4) -> U131#(X1,X2,X3,X4)
p2: U131#(X1,X2,active(X3),X4) -> U131#(X1,X2,X3,X4)
p3: U131#(X1,active(X2),X3,X4) -> U131#(X1,X2,X3,X4)
p4: U131#(X1,X2,mark(X3),X4) -> U131#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U131#_A(x1,x2,x3,x4) = max{0, x3 - 1}
    mark_A(x1) = x1 + 2
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p4

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U131#(X1,mark(X2),X3,X4) -> U131#(X1,X2,X3,X4)
p2: U131#(X1,X2,active(X3),X4) -> U131#(X1,X2,X3,X4)
p3: U131#(X1,active(X2),X3,X4) -> U131#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2, p3}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U131#(X1,mark(X2),X3,X4) -> U131#(X1,X2,X3,X4)
p2: U131#(X1,active(X2),X3,X4) -> U131#(X1,X2,X3,X4)
p3: U131#(X1,X2,active(X3),X4) -> U131#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U131#_A(x1,x2,x3,x4) = x2 + 1
    mark_A(x1) = x1
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U131#(X1,mark(X2),X3,X4) -> U131#(X1,X2,X3,X4)
p2: U131#(X1,X2,active(X3),X4) -> U131#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1, p2}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U131#(X1,mark(X2),X3,X4) -> U131#(X1,X2,X3,X4)
p2: U131#(X1,X2,active(X3),X4) -> U131#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U131#_A(x1,x2,x3,x4) = x3
    mark_A(x1) = max{0, x1 - 1}
    active_A(x1) = x1 + 1

The next rules are strictly ordered:

  p2

We remove them from the problem.

-- SCC decomposition.

Consider the dependency pair problem (P, R), where P consists of

p1: U131#(X1,mark(X2),X3,X4) -> U131#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The estimated dependency graph contains the following SCCs:

  {p1}


-- Reduction pair.

Consider the dependency pair problem (P, R), where P consists of

p1: U131#(X1,mark(X2),X3,X4) -> U131#(X1,X2,X3,X4)

and R consists of:

r1: active(zeros()) -> mark(cons(|0|(),zeros()))
r2: active(U101(tt(),V1,V2)) -> mark(U102(isNatKind(V1),V1,V2))
r3: active(U102(tt(),V1,V2)) -> mark(U103(isNatIListKind(V2),V1,V2))
r4: active(U103(tt(),V1,V2)) -> mark(U104(isNatIListKind(V2),V1,V2))
r5: active(U104(tt(),V1,V2)) -> mark(U105(isNat(V1),V2))
r6: active(U105(tt(),V2)) -> mark(U106(isNatIList(V2)))
r7: active(U106(tt())) -> mark(tt())
r8: active(U11(tt(),V1)) -> mark(U12(isNatIListKind(V1),V1))
r9: active(U111(tt(),L,N)) -> mark(U112(isNatIListKind(L),L,N))
r10: active(U112(tt(),L,N)) -> mark(U113(isNat(N),L,N))
r11: active(U113(tt(),L,N)) -> mark(U114(isNatKind(N),L))
r12: active(U114(tt(),L)) -> mark(s(length(L)))
r13: active(U12(tt(),V1)) -> mark(U13(isNatList(V1)))
r14: active(U121(tt(),IL)) -> mark(U122(isNatIListKind(IL)))
r15: active(U122(tt())) -> mark(nil())
r16: active(U13(tt())) -> mark(tt())
r17: active(U131(tt(),IL,M,N)) -> mark(U132(isNatIListKind(IL),IL,M,N))
r18: active(U132(tt(),IL,M,N)) -> mark(U133(isNat(M),IL,M,N))
r19: active(U133(tt(),IL,M,N)) -> mark(U134(isNatKind(M),IL,M,N))
r20: active(U134(tt(),IL,M,N)) -> mark(U135(isNat(N),IL,M,N))
r21: active(U135(tt(),IL,M,N)) -> mark(U136(isNatKind(N),IL,M,N))
r22: active(U136(tt(),IL,M,N)) -> mark(cons(N,take(M,IL)))
r23: active(U21(tt(),V1)) -> mark(U22(isNatKind(V1),V1))
r24: active(U22(tt(),V1)) -> mark(U23(isNat(V1)))
r25: active(U23(tt())) -> mark(tt())
r26: active(U31(tt(),V)) -> mark(U32(isNatIListKind(V),V))
r27: active(U32(tt(),V)) -> mark(U33(isNatList(V)))
r28: active(U33(tt())) -> mark(tt())
r29: active(U41(tt(),V1,V2)) -> mark(U42(isNatKind(V1),V1,V2))
r30: active(U42(tt(),V1,V2)) -> mark(U43(isNatIListKind(V2),V1,V2))
r31: active(U43(tt(),V1,V2)) -> mark(U44(isNatIListKind(V2),V1,V2))
r32: active(U44(tt(),V1,V2)) -> mark(U45(isNat(V1),V2))
r33: active(U45(tt(),V2)) -> mark(U46(isNatIList(V2)))
r34: active(U46(tt())) -> mark(tt())
r35: active(U51(tt(),V2)) -> mark(U52(isNatIListKind(V2)))
r36: active(U52(tt())) -> mark(tt())
r37: active(U61(tt(),V2)) -> mark(U62(isNatIListKind(V2)))
r38: active(U62(tt())) -> mark(tt())
r39: active(U71(tt())) -> mark(tt())
r40: active(U81(tt())) -> mark(tt())
r41: active(U91(tt(),V1,V2)) -> mark(U92(isNatKind(V1),V1,V2))
r42: active(U92(tt(),V1,V2)) -> mark(U93(isNatIListKind(V2),V1,V2))
r43: active(U93(tt(),V1,V2)) -> mark(U94(isNatIListKind(V2),V1,V2))
r44: active(U94(tt(),V1,V2)) -> mark(U95(isNat(V1),V2))
r45: active(U95(tt(),V2)) -> mark(U96(isNatList(V2)))
r46: active(U96(tt())) -> mark(tt())
r47: active(isNat(|0|())) -> mark(tt())
r48: active(isNat(length(V1))) -> mark(U11(isNatIListKind(V1),V1))
r49: active(isNat(s(V1))) -> mark(U21(isNatKind(V1),V1))
r50: active(isNatIList(V)) -> mark(U31(isNatIListKind(V),V))
r51: active(isNatIList(zeros())) -> mark(tt())
r52: active(isNatIList(cons(V1,V2))) -> mark(U41(isNatKind(V1),V1,V2))
r53: active(isNatIListKind(nil())) -> mark(tt())
r54: active(isNatIListKind(zeros())) -> mark(tt())
r55: active(isNatIListKind(cons(V1,V2))) -> mark(U51(isNatKind(V1),V2))
r56: active(isNatIListKind(take(V1,V2))) -> mark(U61(isNatKind(V1),V2))
r57: active(isNatKind(|0|())) -> mark(tt())
r58: active(isNatKind(length(V1))) -> mark(U71(isNatIListKind(V1)))
r59: active(isNatKind(s(V1))) -> mark(U81(isNatKind(V1)))
r60: active(isNatList(nil())) -> mark(tt())
r61: active(isNatList(cons(V1,V2))) -> mark(U91(isNatKind(V1),V1,V2))
r62: active(isNatList(take(V1,V2))) -> mark(U101(isNatKind(V1),V1,V2))
r63: active(length(nil())) -> mark(|0|())
r64: active(length(cons(N,L))) -> mark(U111(isNatList(L),L,N))
r65: active(take(|0|(),IL)) -> mark(U121(isNatIList(IL),IL))
r66: active(take(s(M),cons(N,IL))) -> mark(U131(isNatIList(IL),IL,M,N))
r67: mark(zeros()) -> active(zeros())
r68: mark(cons(X1,X2)) -> active(cons(mark(X1),X2))
r69: mark(|0|()) -> active(|0|())
r70: mark(U101(X1,X2,X3)) -> active(U101(mark(X1),X2,X3))
r71: mark(tt()) -> active(tt())
r72: mark(U102(X1,X2,X3)) -> active(U102(mark(X1),X2,X3))
r73: mark(isNatKind(X)) -> active(isNatKind(X))
r74: mark(U103(X1,X2,X3)) -> active(U103(mark(X1),X2,X3))
r75: mark(isNatIListKind(X)) -> active(isNatIListKind(X))
r76: mark(U104(X1,X2,X3)) -> active(U104(mark(X1),X2,X3))
r77: mark(U105(X1,X2)) -> active(U105(mark(X1),X2))
r78: mark(isNat(X)) -> active(isNat(X))
r79: mark(U106(X)) -> active(U106(mark(X)))
r80: mark(isNatIList(X)) -> active(isNatIList(X))
r81: mark(U11(X1,X2)) -> active(U11(mark(X1),X2))
r82: mark(U12(X1,X2)) -> active(U12(mark(X1),X2))
r83: mark(U111(X1,X2,X3)) -> active(U111(mark(X1),X2,X3))
r84: mark(U112(X1,X2,X3)) -> active(U112(mark(X1),X2,X3))
r85: mark(U113(X1,X2,X3)) -> active(U113(mark(X1),X2,X3))
r86: mark(U114(X1,X2)) -> active(U114(mark(X1),X2))
r87: mark(s(X)) -> active(s(mark(X)))
r88: mark(length(X)) -> active(length(mark(X)))
r89: mark(U13(X)) -> active(U13(mark(X)))
r90: mark(isNatList(X)) -> active(isNatList(X))
r91: mark(U121(X1,X2)) -> active(U121(mark(X1),X2))
r92: mark(U122(X)) -> active(U122(mark(X)))
r93: mark(nil()) -> active(nil())
r94: mark(U131(X1,X2,X3,X4)) -> active(U131(mark(X1),X2,X3,X4))
r95: mark(U132(X1,X2,X3,X4)) -> active(U132(mark(X1),X2,X3,X4))
r96: mark(U133(X1,X2,X3,X4)) -> active(U133(mark(X1),X2,X3,X4))
r97: mark(U134(X1,X2,X3,X4)) -> active(U134(mark(X1),X2,X3,X4))
r98: mark(U135(X1,X2,X3,X4)) -> active(U135(mark(X1),X2,X3,X4))
r99: mark(U136(X1,X2,X3,X4)) -> active(U136(mark(X1),X2,X3,X4))
r100: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2)))
r101: mark(U21(X1,X2)) -> active(U21(mark(X1),X2))
r102: mark(U22(X1,X2)) -> active(U22(mark(X1),X2))
r103: mark(U23(X)) -> active(U23(mark(X)))
r104: mark(U31(X1,X2)) -> active(U31(mark(X1),X2))
r105: mark(U32(X1,X2)) -> active(U32(mark(X1),X2))
r106: mark(U33(X)) -> active(U33(mark(X)))
r107: mark(U41(X1,X2,X3)) -> active(U41(mark(X1),X2,X3))
r108: mark(U42(X1,X2,X3)) -> active(U42(mark(X1),X2,X3))
r109: mark(U43(X1,X2,X3)) -> active(U43(mark(X1),X2,X3))
r110: mark(U44(X1,X2,X3)) -> active(U44(mark(X1),X2,X3))
r111: mark(U45(X1,X2)) -> active(U45(mark(X1),X2))
r112: mark(U46(X)) -> active(U46(mark(X)))
r113: mark(U51(X1,X2)) -> active(U51(mark(X1),X2))
r114: mark(U52(X)) -> active(U52(mark(X)))
r115: mark(U61(X1,X2)) -> active(U61(mark(X1),X2))
r116: mark(U62(X)) -> active(U62(mark(X)))
r117: mark(U71(X)) -> active(U71(mark(X)))
r118: mark(U81(X)) -> active(U81(mark(X)))
r119: mark(U91(X1,X2,X3)) -> active(U91(mark(X1),X2,X3))
r120: mark(U92(X1,X2,X3)) -> active(U92(mark(X1),X2,X3))
r121: mark(U93(X1,X2,X3)) -> active(U93(mark(X1),X2,X3))
r122: mark(U94(X1,X2,X3)) -> active(U94(mark(X1),X2,X3))
r123: mark(U95(X1,X2)) -> active(U95(mark(X1),X2))
r124: mark(U96(X)) -> active(U96(mark(X)))
r125: cons(mark(X1),X2) -> cons(X1,X2)
r126: cons(X1,mark(X2)) -> cons(X1,X2)
r127: cons(active(X1),X2) -> cons(X1,X2)
r128: cons(X1,active(X2)) -> cons(X1,X2)
r129: U101(mark(X1),X2,X3) -> U101(X1,X2,X3)
r130: U101(X1,mark(X2),X3) -> U101(X1,X2,X3)
r131: U101(X1,X2,mark(X3)) -> U101(X1,X2,X3)
r132: U101(active(X1),X2,X3) -> U101(X1,X2,X3)
r133: U101(X1,active(X2),X3) -> U101(X1,X2,X3)
r134: U101(X1,X2,active(X3)) -> U101(X1,X2,X3)
r135: U102(mark(X1),X2,X3) -> U102(X1,X2,X3)
r136: U102(X1,mark(X2),X3) -> U102(X1,X2,X3)
r137: U102(X1,X2,mark(X3)) -> U102(X1,X2,X3)
r138: U102(active(X1),X2,X3) -> U102(X1,X2,X3)
r139: U102(X1,active(X2),X3) -> U102(X1,X2,X3)
r140: U102(X1,X2,active(X3)) -> U102(X1,X2,X3)
r141: isNatKind(mark(X)) -> isNatKind(X)
r142: isNatKind(active(X)) -> isNatKind(X)
r143: U103(mark(X1),X2,X3) -> U103(X1,X2,X3)
r144: U103(X1,mark(X2),X3) -> U103(X1,X2,X3)
r145: U103(X1,X2,mark(X3)) -> U103(X1,X2,X3)
r146: U103(active(X1),X2,X3) -> U103(X1,X2,X3)
r147: U103(X1,active(X2),X3) -> U103(X1,X2,X3)
r148: U103(X1,X2,active(X3)) -> U103(X1,X2,X3)
r149: isNatIListKind(mark(X)) -> isNatIListKind(X)
r150: isNatIListKind(active(X)) -> isNatIListKind(X)
r151: U104(mark(X1),X2,X3) -> U104(X1,X2,X3)
r152: U104(X1,mark(X2),X3) -> U104(X1,X2,X3)
r153: U104(X1,X2,mark(X3)) -> U104(X1,X2,X3)
r154: U104(active(X1),X2,X3) -> U104(X1,X2,X3)
r155: U104(X1,active(X2),X3) -> U104(X1,X2,X3)
r156: U104(X1,X2,active(X3)) -> U104(X1,X2,X3)
r157: U105(mark(X1),X2) -> U105(X1,X2)
r158: U105(X1,mark(X2)) -> U105(X1,X2)
r159: U105(active(X1),X2) -> U105(X1,X2)
r160: U105(X1,active(X2)) -> U105(X1,X2)
r161: isNat(mark(X)) -> isNat(X)
r162: isNat(active(X)) -> isNat(X)
r163: U106(mark(X)) -> U106(X)
r164: U106(active(X)) -> U106(X)
r165: isNatIList(mark(X)) -> isNatIList(X)
r166: isNatIList(active(X)) -> isNatIList(X)
r167: U11(mark(X1),X2) -> U11(X1,X2)
r168: U11(X1,mark(X2)) -> U11(X1,X2)
r169: U11(active(X1),X2) -> U11(X1,X2)
r170: U11(X1,active(X2)) -> U11(X1,X2)
r171: U12(mark(X1),X2) -> U12(X1,X2)
r172: U12(X1,mark(X2)) -> U12(X1,X2)
r173: U12(active(X1),X2) -> U12(X1,X2)
r174: U12(X1,active(X2)) -> U12(X1,X2)
r175: U111(mark(X1),X2,X3) -> U111(X1,X2,X3)
r176: U111(X1,mark(X2),X3) -> U111(X1,X2,X3)
r177: U111(X1,X2,mark(X3)) -> U111(X1,X2,X3)
r178: U111(active(X1),X2,X3) -> U111(X1,X2,X3)
r179: U111(X1,active(X2),X3) -> U111(X1,X2,X3)
r180: U111(X1,X2,active(X3)) -> U111(X1,X2,X3)
r181: U112(mark(X1),X2,X3) -> U112(X1,X2,X3)
r182: U112(X1,mark(X2),X3) -> U112(X1,X2,X3)
r183: U112(X1,X2,mark(X3)) -> U112(X1,X2,X3)
r184: U112(active(X1),X2,X3) -> U112(X1,X2,X3)
r185: U112(X1,active(X2),X3) -> U112(X1,X2,X3)
r186: U112(X1,X2,active(X3)) -> U112(X1,X2,X3)
r187: U113(mark(X1),X2,X3) -> U113(X1,X2,X3)
r188: U113(X1,mark(X2),X3) -> U113(X1,X2,X3)
r189: U113(X1,X2,mark(X3)) -> U113(X1,X2,X3)
r190: U113(active(X1),X2,X3) -> U113(X1,X2,X3)
r191: U113(X1,active(X2),X3) -> U113(X1,X2,X3)
r192: U113(X1,X2,active(X3)) -> U113(X1,X2,X3)
r193: U114(mark(X1),X2) -> U114(X1,X2)
r194: U114(X1,mark(X2)) -> U114(X1,X2)
r195: U114(active(X1),X2) -> U114(X1,X2)
r196: U114(X1,active(X2)) -> U114(X1,X2)
r197: s(mark(X)) -> s(X)
r198: s(active(X)) -> s(X)
r199: length(mark(X)) -> length(X)
r200: length(active(X)) -> length(X)
r201: U13(mark(X)) -> U13(X)
r202: U13(active(X)) -> U13(X)
r203: isNatList(mark(X)) -> isNatList(X)
r204: isNatList(active(X)) -> isNatList(X)
r205: U121(mark(X1),X2) -> U121(X1,X2)
r206: U121(X1,mark(X2)) -> U121(X1,X2)
r207: U121(active(X1),X2) -> U121(X1,X2)
r208: U121(X1,active(X2)) -> U121(X1,X2)
r209: U122(mark(X)) -> U122(X)
r210: U122(active(X)) -> U122(X)
r211: U131(mark(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r212: U131(X1,mark(X2),X3,X4) -> U131(X1,X2,X3,X4)
r213: U131(X1,X2,mark(X3),X4) -> U131(X1,X2,X3,X4)
r214: U131(X1,X2,X3,mark(X4)) -> U131(X1,X2,X3,X4)
r215: U131(active(X1),X2,X3,X4) -> U131(X1,X2,X3,X4)
r216: U131(X1,active(X2),X3,X4) -> U131(X1,X2,X3,X4)
r217: U131(X1,X2,active(X3),X4) -> U131(X1,X2,X3,X4)
r218: U131(X1,X2,X3,active(X4)) -> U131(X1,X2,X3,X4)
r219: U132(mark(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r220: U132(X1,mark(X2),X3,X4) -> U132(X1,X2,X3,X4)
r221: U132(X1,X2,mark(X3),X4) -> U132(X1,X2,X3,X4)
r222: U132(X1,X2,X3,mark(X4)) -> U132(X1,X2,X3,X4)
r223: U132(active(X1),X2,X3,X4) -> U132(X1,X2,X3,X4)
r224: U132(X1,active(X2),X3,X4) -> U132(X1,X2,X3,X4)
r225: U132(X1,X2,active(X3),X4) -> U132(X1,X2,X3,X4)
r226: U132(X1,X2,X3,active(X4)) -> U132(X1,X2,X3,X4)
r227: U133(mark(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r228: U133(X1,mark(X2),X3,X4) -> U133(X1,X2,X3,X4)
r229: U133(X1,X2,mark(X3),X4) -> U133(X1,X2,X3,X4)
r230: U133(X1,X2,X3,mark(X4)) -> U133(X1,X2,X3,X4)
r231: U133(active(X1),X2,X3,X4) -> U133(X1,X2,X3,X4)
r232: U133(X1,active(X2),X3,X4) -> U133(X1,X2,X3,X4)
r233: U133(X1,X2,active(X3),X4) -> U133(X1,X2,X3,X4)
r234: U133(X1,X2,X3,active(X4)) -> U133(X1,X2,X3,X4)
r235: U134(mark(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r236: U134(X1,mark(X2),X3,X4) -> U134(X1,X2,X3,X4)
r237: U134(X1,X2,mark(X3),X4) -> U134(X1,X2,X3,X4)
r238: U134(X1,X2,X3,mark(X4)) -> U134(X1,X2,X3,X4)
r239: U134(active(X1),X2,X3,X4) -> U134(X1,X2,X3,X4)
r240: U134(X1,active(X2),X3,X4) -> U134(X1,X2,X3,X4)
r241: U134(X1,X2,active(X3),X4) -> U134(X1,X2,X3,X4)
r242: U134(X1,X2,X3,active(X4)) -> U134(X1,X2,X3,X4)
r243: U135(mark(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r244: U135(X1,mark(X2),X3,X4) -> U135(X1,X2,X3,X4)
r245: U135(X1,X2,mark(X3),X4) -> U135(X1,X2,X3,X4)
r246: U135(X1,X2,X3,mark(X4)) -> U135(X1,X2,X3,X4)
r247: U135(active(X1),X2,X3,X4) -> U135(X1,X2,X3,X4)
r248: U135(X1,active(X2),X3,X4) -> U135(X1,X2,X3,X4)
r249: U135(X1,X2,active(X3),X4) -> U135(X1,X2,X3,X4)
r250: U135(X1,X2,X3,active(X4)) -> U135(X1,X2,X3,X4)
r251: U136(mark(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r252: U136(X1,mark(X2),X3,X4) -> U136(X1,X2,X3,X4)
r253: U136(X1,X2,mark(X3),X4) -> U136(X1,X2,X3,X4)
r254: U136(X1,X2,X3,mark(X4)) -> U136(X1,X2,X3,X4)
r255: U136(active(X1),X2,X3,X4) -> U136(X1,X2,X3,X4)
r256: U136(X1,active(X2),X3,X4) -> U136(X1,X2,X3,X4)
r257: U136(X1,X2,active(X3),X4) -> U136(X1,X2,X3,X4)
r258: U136(X1,X2,X3,active(X4)) -> U136(X1,X2,X3,X4)
r259: take(mark(X1),X2) -> take(X1,X2)
r260: take(X1,mark(X2)) -> take(X1,X2)
r261: take(active(X1),X2) -> take(X1,X2)
r262: take(X1,active(X2)) -> take(X1,X2)
r263: U21(mark(X1),X2) -> U21(X1,X2)
r264: U21(X1,mark(X2)) -> U21(X1,X2)
r265: U21(active(X1),X2) -> U21(X1,X2)
r266: U21(X1,active(X2)) -> U21(X1,X2)
r267: U22(mark(X1),X2) -> U22(X1,X2)
r268: U22(X1,mark(X2)) -> U22(X1,X2)
r269: U22(active(X1),X2) -> U22(X1,X2)
r270: U22(X1,active(X2)) -> U22(X1,X2)
r271: U23(mark(X)) -> U23(X)
r272: U23(active(X)) -> U23(X)
r273: U31(mark(X1),X2) -> U31(X1,X2)
r274: U31(X1,mark(X2)) -> U31(X1,X2)
r275: U31(active(X1),X2) -> U31(X1,X2)
r276: U31(X1,active(X2)) -> U31(X1,X2)
r277: U32(mark(X1),X2) -> U32(X1,X2)
r278: U32(X1,mark(X2)) -> U32(X1,X2)
r279: U32(active(X1),X2) -> U32(X1,X2)
r280: U32(X1,active(X2)) -> U32(X1,X2)
r281: U33(mark(X)) -> U33(X)
r282: U33(active(X)) -> U33(X)
r283: U41(mark(X1),X2,X3) -> U41(X1,X2,X3)
r284: U41(X1,mark(X2),X3) -> U41(X1,X2,X3)
r285: U41(X1,X2,mark(X3)) -> U41(X1,X2,X3)
r286: U41(active(X1),X2,X3) -> U41(X1,X2,X3)
r287: U41(X1,active(X2),X3) -> U41(X1,X2,X3)
r288: U41(X1,X2,active(X3)) -> U41(X1,X2,X3)
r289: U42(mark(X1),X2,X3) -> U42(X1,X2,X3)
r290: U42(X1,mark(X2),X3) -> U42(X1,X2,X3)
r291: U42(X1,X2,mark(X3)) -> U42(X1,X2,X3)
r292: U42(active(X1),X2,X3) -> U42(X1,X2,X3)
r293: U42(X1,active(X2),X3) -> U42(X1,X2,X3)
r294: U42(X1,X2,active(X3)) -> U42(X1,X2,X3)
r295: U43(mark(X1),X2,X3) -> U43(X1,X2,X3)
r296: U43(X1,mark(X2),X3) -> U43(X1,X2,X3)
r297: U43(X1,X2,mark(X3)) -> U43(X1,X2,X3)
r298: U43(active(X1),X2,X3) -> U43(X1,X2,X3)
r299: U43(X1,active(X2),X3) -> U43(X1,X2,X3)
r300: U43(X1,X2,active(X3)) -> U43(X1,X2,X3)
r301: U44(mark(X1),X2,X3) -> U44(X1,X2,X3)
r302: U44(X1,mark(X2),X3) -> U44(X1,X2,X3)
r303: U44(X1,X2,mark(X3)) -> U44(X1,X2,X3)
r304: U44(active(X1),X2,X3) -> U44(X1,X2,X3)
r305: U44(X1,active(X2),X3) -> U44(X1,X2,X3)
r306: U44(X1,X2,active(X3)) -> U44(X1,X2,X3)
r307: U45(mark(X1),X2) -> U45(X1,X2)
r308: U45(X1,mark(X2)) -> U45(X1,X2)
r309: U45(active(X1),X2) -> U45(X1,X2)
r310: U45(X1,active(X2)) -> U45(X1,X2)
r311: U46(mark(X)) -> U46(X)
r312: U46(active(X)) -> U46(X)
r313: U51(mark(X1),X2) -> U51(X1,X2)
r314: U51(X1,mark(X2)) -> U51(X1,X2)
r315: U51(active(X1),X2) -> U51(X1,X2)
r316: U51(X1,active(X2)) -> U51(X1,X2)
r317: U52(mark(X)) -> U52(X)
r318: U52(active(X)) -> U52(X)
r319: U61(mark(X1),X2) -> U61(X1,X2)
r320: U61(X1,mark(X2)) -> U61(X1,X2)
r321: U61(active(X1),X2) -> U61(X1,X2)
r322: U61(X1,active(X2)) -> U61(X1,X2)
r323: U62(mark(X)) -> U62(X)
r324: U62(active(X)) -> U62(X)
r325: U71(mark(X)) -> U71(X)
r326: U71(active(X)) -> U71(X)
r327: U81(mark(X)) -> U81(X)
r328: U81(active(X)) -> U81(X)
r329: U91(mark(X1),X2,X3) -> U91(X1,X2,X3)
r330: U91(X1,mark(X2),X3) -> U91(X1,X2,X3)
r331: U91(X1,X2,mark(X3)) -> U91(X1,X2,X3)
r332: U91(active(X1),X2,X3) -> U91(X1,X2,X3)
r333: U91(X1,active(X2),X3) -> U91(X1,X2,X3)
r334: U91(X1,X2,active(X3)) -> U91(X1,X2,X3)
r335: U92(mark(X1),X2,X3) -> U92(X1,X2,X3)
r336: U92(X1,mark(X2),X3) -> U92(X1,X2,X3)
r337: U92(X1,X2,mark(X3)) -> U92(X1,X2,X3)
r338: U92(active(X1),X2,X3) -> U92(X1,X2,X3)
r339: U92(X1,active(X2),X3) -> U92(X1,X2,X3)
r340: U92(X1,X2,active(X3)) -> U92(X1,X2,X3)
r341: U93(mark(X1),X2,X3) -> U93(X1,X2,X3)
r342: U93(X1,mark(X2),X3) -> U93(X1,X2,X3)
r343: U93(X1,X2,mark(X3)) -> U93(X1,X2,X3)
r344: U93(active(X1),X2,X3) -> U93(X1,X2,X3)
r345: U93(X1,active(X2),X3) -> U93(X1,X2,X3)
r346: U93(X1,X2,active(X3)) -> U93(X1,X2,X3)
r347: U94(mark(X1),X2,X3) -> U94(X1,X2,X3)
r348: U94(X1,mark(X2),X3) -> U94(X1,X2,X3)
r349: U94(X1,X2,mark(X3)) -> U94(X1,X2,X3)
r350: U94(active(X1),X2,X3) -> U94(X1,X2,X3)
r351: U94(X1,active(X2),X3) -> U94(X1,X2,X3)
r352: U94(X1,X2,active(X3)) -> U94(X1,X2,X3)
r353: U95(mark(X1),X2) -> U95(X1,X2)
r354: U95(X1,mark(X2)) -> U95(X1,X2)
r355: U95(active(X1),X2) -> U95(X1,X2)
r356: U95(X1,active(X2)) -> U95(X1,X2)
r357: U96(mark(X)) -> U96(X)
r358: U96(active(X)) -> U96(X)

The set of usable rules consists of

  (no rules)

Take the reduction pair:

  max/plus interpretations on natural numbers:
  
    U131#_A(x1,x2,x3,x4) = x2
    mark_A(x1) = x1 + 1

The next rules are strictly ordered:

  p1

We remove them from the problem.  Then no dependency pair remains.